PATH SPACES, CONTINUOUS TENSOR

PRODUCTS, AND E_{0}-SEMIGROUPS

William Arveson

Department of Mathematics University of California Berkeley CA 94720, USA

Abstract. We classify all continuous tensor product systems of Hilbert spaces which are “infinitely divisible” in the sense that they have an associated logarithmic structure. These results are applied to the theory of E_{0}-semigroups to deduce that every E_{0}-semigroup possessing sufficiently many “decomposable” operators must be cocycle conjugate to a CCR flow.

A path space is an abstraction of the set of paths in a topological space, on which there is given an associative rule of concatenation. A metric path space is a pair (P, g) consisting of a path space P and a function g : P^{2} → ℂ which behaves as if it were the logarithm of a multiplicative inner product. The logarithmic structures associated with infinitely divisible product systems are such objects. The preceding results are based on a classification of metric path spaces.

Contents

Introduction Part I. Path Spaces

1. Definitions and examples

2. Additive forms and multiplicative forms

3. Exactness of cocycles

4. Classification of additive forms Part II. Continuous Tensor Products

5. Continuity of the modulus

6. Inner products of decomposable vectors

7. Continuity and normalization

8. Continuous logarithms

9. Infinite divisibility

10. Existence of measurable propagators Part III. Applications

11. Decomposable continuous tensor products

12. Decomposable E_{0}-semigroups

1991 Mathematics Subject Classification. Primary 46L40; Secondary 81E05. Key words and phrases. von Neumann algebras, semigroups, automorphism groups. This research was supported in part by NSF grant DMS92-43893

Typeset by AMS-TEX

Introduction. Let α = {α_{t} : t > 0} be an E_{0}-semigroup. That is, each α_{t} is a normal unit-preserving ∗-endomorphism of B(H) such that α_{s} ∘ α_{t} = α_{s+t}, and which is continuous in the sense that for all A ∈ B(H), ξ, η ∈ H, ⟨α_{t}(A)ξ, η⟩ is continuous in t.

A unit for α is a strongly continuous semigroup U = {U(t) : t ≥ 0} of bounded operators on H satisfying

α_{t}(A)U(t) = U(t)A,

for every t ≥ 0, A ∈ B(H). α is called completely spatial if there is a t > 0 such that H is spanned by the ranges of all operator products of the form

U_{1}(t_{1})U_{2}(t_{2}) … U_{n}(t_{n})

where U_{1}, U_{2}, …, U_{n} are units, t_{1}, t_{2}, …, t_{n} are nonnegative reals summing to t, and n is an arbitrary positive integer. Completely spatial E_{0}-semigroups are those which have “sufficiently many” units.

In [2, section 7], completely spatial E_{0}-semigroups are shown to be completely classified up to cocycle conjugacy by their numerical index. This is established at the level of continuous tensor product systems. In somewhat more detail, a product system is a measurable family of separable Hilbert spaces E = {E_{t} : t > 0} which is endowed with an associative multiplication that “acts like tensoring” in the sense that for any choices x, x′ ∈ E_{s}, y, y′ ∈ E_{t}, the products xy, x′y′ both belong to E_{s+t} and we have

⟨xy, x′y′⟩ = ⟨x, x′⟩ ⟨y, y′⟩ ,  and
E_{s+t} = \overline{span}E_{s}E_{t}.

In addition, there is a third key axiom which plays the role of local triviality for Hermitian vector bundles. The precise definition of product systems can be found in [2, Definition 1.4]. The intuition is that E_{t} resembles a continuous tensor product

(I.1)  E_{t} = ⨂_{0<s<t} H_{s},  H_{s} = H

of copies of a single Hilbert space H. However, this heuristic picture of continuous tensor products cannot be pushed too far. While the formula (I.1) can be made precise for certain standard examples, it is also known that there are many product systems for which the “germ” H fails to exist.

Every E_{0}-semigroup α determines a product system E_{α}. The t^{th} Hilbert space E_{α}(t) is the linear space of operators

E_{α}(t) = {T ∈ B(H) : α_{t}(A)T = T A, ∀A ∈ B(H)},

with inner product defined by

⟨S, T ⟩ 1 = T^{∗}S, S, T ∈ E_{α}(t),

and with multiplication given by ordinary operator multiplication. It is known that two E_{0}-semigroups are cocycle conjugate iff their product systems E_{α}, E_{β} are isomorphic [2, Corollary of Theorem 3.18]; moreover, every abstract product system is associated with some E_{0}-semigroup [5, Corollary 5.17].

These remarks show that, up to cocycle conjugacy, the theory of E_{0}-semigroups is equivalent to the theory of continuous tensor product systems. Consequently, a central component of our approach to E_{0}-semigroups has been to develop the theory of product systems. The classification of E_{0}-semigroups described above was obtained by classifying product systems which possess a property that corresponds to complete spatiality of their product systems (this was called divisibility in [2]).

In this paper, we extend that result so as to include certain product systems which do not appear to contain any units a priori, but which do contain sufficiently many vectors that resemble “elementary tensors”. Such product systems are called decomposable. The corresponding property of E_{0}-semigroups is described as follows. Fix t > 0. An operator T ∈ E_{α}(t) is called decomposable if, for every 0 < s < t there are operators A ∈ E_{α}(s), B ∈ E_{α}(t − s) such that T = AB. Let D(t) denote the set of all decomposable operators in E_{α}(t). It is easy to see that if H is spanned by vectors of the form

D(t_{0})H = {T ξ : T ∈ D(t_{0}), ξ ∈ H}

for some particular t_{0} > 0, then we have

(I.2)  H = [D(t)H]

for every t > 0. α is called decomposable if (I.2) is valid for some t > 0. Notice that if U_{1}, U_{2}, …, U_{n} are units for α and t_{1}, t_{2}, …, t_{n} are positive numbers summing to t, then every operator product of the form

U_{1}(t_{1})U_{2}(t_{2}) … U_{n}(t_{n})

is a decomposable operator in E_{α}(t) because each U_{j} is a semigroup. Thus, every completely spatial E_{0}-semigroup is decomposable.

On the other hand, decomposable E_{0}-semigroups are not required to contain any units a priori. The results of this paper imply that decomposable E_{0}-semigroups are completely spatial, and are therefore classified to cocycle conjugacy by their numerical index. In particular, decomposable E_{0}-semigroups necessarily have plenty of units.

Decomposability translates into an important property of abstract product systems, and we want to discuss this property and its role in structural issues. Let E = {E(t) : t > 0} be an abstract product system and fix t > 0. A nonzero vector x ∈ E(t) is called decomposable if for every s ∈ (0, t) there are vectors a ∈ E(s), b ∈ E(t − s) such that x = ab. Let D(t) be the set of all decomposable vectors in E(t). D(t) can be empty. But if it is large enough that it spans E(t) for some particular t, then it is easy to show that

(I.3)  E(t) = \overline{span}D(t)

for every positive t. A product system satisfying (I.3) for every positive t is called decomposable. It is also easy to show that if E is isomorphic to the product system E_{α} of some E_{0}-semigroup α, then E is decomposable iff α is (Proposition 12.2).

If we think of the multiplication in a product system E as representing the tensor product operation, then the vectors in D(t) are “elementary tensors”. In the heuristic picture in which one thinks of E(t) as a continuous tensor product

E(t) = ⨂_{0<s<t} H_{s}, H_{s} = H

of copies of a Hilbert space H, the decomposable vectors have a corresponding heuristic representation

(I.4)  x = ⊗_{0<s<t} x_{s}

where x_{s} is a nonzero vector in H for every s ∈ (0, t). Since finite tensor products of Hilbert spaces of the form

\underbrace{H ⊗ H ⊗ ⋯ ⊗ H}_{n times}

are always spanned by vectors of the form

x = x_{1} ⊗ x_{2} ⊗ ⋯ ⊗ x_{n}

with x_{i} ∈ H, decomposability appears as a natural property of product systems. However, notice that Theorem 11.1 below implies that only the simplest product systems are decomposable.

Product systems occur naturally in probability theory. For example, any random distribution over the interval (0, ∞) having stationary independent increments gives rise to a product system (see [2, pp 14–16] for an example based on Poisson processes). Some of these examples (such those arising from Poisson processes) do not appear to have “enough” units to be divisible in the sense of [2]. Nevertheless, they do. They do because in constructing these examples, one essentially constructs the sets D(t), t > 0, and then defines the product system E by setting

E(t) = \overline{span}D(t), t > 0.

Such a product system is clearly decomposable and thus, by Theorem 11.1 below, is simply one of the standard ones.

Our approach to decomposable product systems is based on the concept of a metric path space. Roughly speaking, a metric path space is a structure which behaves as if it were the “logarithm” of a product system. Path spaces are very general objects, and we give a variety of examples. The most natural constructions of product systems all begin with a metric path space of some kind...an example of this being the construction starting with a Poisson process as above. We give an essentially complete classification of metric path spaces in part I. In part II, we show that every decomposable product system is associated with an essentially unique metric path space, and in part III these results are applied to the classification of product systems and E_{0}-semigroups.

We still do not know a “natural” construction of a product system without any units, or even one with some units but not one of the standard ones. Such product systems are known to exist by work of Powers [25, 27] together with the results of [2]. Powers’ constructions are very indirect. We believe that there should be a natural way of constructing such product systems, and we offer that as a basic unsolved problem. The fact that we do not yet know how to solve it shows how poorly understood continuous tensor products are today.

It is appropriate to discuss the relation of our results to the work of Araki and Woods on continuous tensor products [1]. Araki and Woods were interested in generalizing results of von Neumann on infinite tensor products of Hilbert spaces [21,22] to the case of continuous tensor products. Their precise formulation of a continuous tensor product of Hilbert spaces is expressed in terms of a nonatomic Boolean algebra of type I factors. Our formulation of continuous tensor products is in terms of the product systems defined above. While both concepts seek to make precise the intuitive idea of continuous tensor products of Hilbert spaces, they are formulated in substantially different and mathematically inequivalent ways.

For example, the Hilbert spaces of [1] are parameterized by the elements of a σ-algebra of sets, whereas the fiber spaces of product systems are parameterized by positive real numbers. If one starts with an abstract product system E then there is no underlying Hilbert space in view. Even if E did act naturally on some Hilbert space (as it does when it is defined in terms of an E_{0}-semigroup), one still cannot use it to define a Boolean algebra of type I factors. It is true that one can associate a type I factor with subintervals of (0, ∞), but one does not know how to extend that set function to arbitrary Borel sets in (0, ∞)...certainly not so that one can be sure the extended set function will take values in the set of type I factors.

Thus, product systems are more general than the structures of Araki-Woods. But even that assertion is not exactly true. We also do not know how to start with a complete Boolean algebra of type I factors over the interval (0, ∞) and use that structure to define a product system. In order to do that, one needs a bit more symmetry with respect to positive translations. Still, I am comfortable with the philosophical position that product systems are generalizations of complete Boolean algebras of type I factors over the interval (0, ∞).

In more pragmatic terms, we have been unable to apply the work of Araki-Woods to the theory of product systems or to the theory of E_{0}-semigroups, even though there are apparent similarities in certain results. For example, Theorem 11.1 below is similar to Theorem 6.1 of [1]. Notice that the hypothesis of the latter, namely

inf_{B} d(Ψ; B) > 0,

corresponds to our hypothesis of decomposability because of Lemma 6.3 of [1]. In fact, in establishing the classification results of [2] as well as those of the present paper, we have had to start from scratch...basing our proofs on different principles than the proofs of corresponding results of [1]. Nevertheless, we certainly acknowledge the considerable influence of [1] in guiding our thinking about continuous tensor products.

Part I: Path Spaces

1. Definitions and examples.

A path space is a family of sets over the positive real axis

π : P → (0, ∞)

which is endowed with an associative multiplication (x, y) ∈ P × P ↦ xy ∈ P satisfying the following two axioms. Letting P (t) = π^{−1}(t) be the fiber over t we require that for every s, t > 0,

(1.1.1)  P (s) · P (t) = P (s + t)

and that for all x_{1}, x_{2} ∈ P (s), y_{1}, y_{2} ∈ P (t)

(1.1.2)  x_{1}y_{1} = x_{2}y_{2} ⟹ x_{1} = x_{2} and y_{1} = y_{2}.

Together, the conditions (1.1) assert that the projection π should obey π(xy) = π(x) + π(y), and that for fixed 0 < s < t, each element z ∈ P (t) factors uniquely into a product z = xy, with x ∈ P (s) and y ∈ P (t − s).

Path spaces are generalizations of continuous cartesian products, whereby one starts with a basic set X and defines fiber spaces P (t) by

(1.2.1)  P (t) = X^{(0,t]}.

P is defined as the total set π : P = {(t, f) : t > 0, f ∈ P (t)} → (0, ∞) with its natural projection π(t, f) = t. The multiplication in P is defined by

(s, f) · (t, g) = (s + t, h)

where h is the concatenation

(1.2.2)  h(x) = \{ \begin{matrix}f(x), & 0 < x ≤ s \\ g(x − s), & s < x ≤ s + t. \end{matrix}

Remarks. The terminology path space suggests that one should think of the elements of P as representing paths in some space. Notice however that if X is a topological space in the preceding example (1.2), the functions in any particular P (t) are not necessarily continuous, nor even measurable. In fact, the examples (1.2) are pathological. We give some more typical examples below. But we emphasize that in general, path spaces are not required to support an additional topological structure, nor even a Borel structure. We will find that, nevertheless, the intrinsic structure of path spaces allows one to define a notion of measurability (see Definition 2.2) and even continuity (see section 8) for the functions defined on them.

Let π : P → (0, ∞) be a path space and let x, y ∈ P. We will say that x is a left divisor of y is there is an element z ∈ P with y = xz. The element z is necessarily unique and, if x ∈ P (s) and y ∈ P (t) then we must have 0 < s < t and z ∈ P (t−s). Fix T > 0. A section

t ∈ (0, T] ↦ x(t) ∈ P (t)

is said to be left-coherent if x(s) is a left divisor of x(t) whenever s < t. Notice that such a section is uniquely determined by its last element x(T) ∈ P (T) because of the unique factorization property. Thus there is a bijective correspondence between elements of P (T) and left-coherent sections t ∈ (0, T] ↦ x(t) ∈ P (t).

There is another description of left coherent sections t ∈ (0, T] ↦ P (t) that we will find useful. Given such a section we can define a family of elements {x(s, t) ∈ P (t − s) : 0 ≤ s < t ≤ T } by

x(t) = x(s)x(s, t), for 0 < s < t ≤ T,

and where x(0, t) is defined as x(t). This family satisfies the consistency equation

(1.3)  0 ≤ r < s < t ≤ T ⟹ x(r, t) = x(r, s)x(s, t).

There is an obvious modification of these considerations for fully defined sections t ∈ (0, ∞) ↦ x(t) ∈ P (t), and in this case we obtain a family {x(s, t) : 0 ≤ s < t < ∞} satisfying (1.3) for unrestricted 0 ≤ r < s < t.

Definition 1.4. Let I be an interval of the form I = [0, T] with 0 < T < ∞ or I = [0, ∞). A family of elements {x(s, t) ∈ P (t − s) : s, t ∈ I, s < t} is called a propagator on I if it satisfies the equation (1.3) for every r, s, t ∈ I such that r < s < t.

Operator-valued functions satisfying equation (1.3) are naturally associated with solutions of time-dependent linear differential equations of the form

\frac{d}{dt}x(t) = x(t)a(t), 0 ≤ t ≤ T,

where a(·) is a given operator-valued function with invertible values [30, p. 282]. For our purposes, propagators are associated with left-coherent sections as above. Notice that the correspondence between propagators and left-coherent sections is also bijective since one can recover the section {x(t)} from the propagator {x(s, t)} by setting

x(t) = x(0, t).

Example 1.5. Let V be a finite dimensional vector space or a Banach space. For every t > 0, let P (t) denote the space of all continuous functions

f : [0, t] → V

satisfying f(0) = 0. For f ∈ P (s), g ∈ P (t) we define a concatenation f ∗g ∈ P (s+t) by

f ∗ g(λ) = \{ \begin{matrix}f(λ), & 0 ≤ λ < s \\ f(s) + g(λ − s), & s ≤ λ ≤ s + t. \end{matrix}

Notice that f ∗ g executes the path f first, and then it executes g. If we assemble these spaces into a family

P = {(t, f) : t > 0, f ∈ P (t)}

with projection π(t, f) = t and multiplication

(s, f) · (t, g) = (s + t, f ∗ g),

then we obtain a path space π : P → (0, ∞).

Example 1.6. One can define variations of example 1.5 with higher degrees of smoothness. For example, let P (t) be the space of all continuously differentiable functions f : [0, t] → V satisfying f(0) = f′(0) = 0. Then with the concatenation rule in which for f ∈ P (s) and g ∈ P (t), f ∗ g is defined by

f ∗ g(λ) = \{ \begin{matrix}f(λ), & 0 ≤ λ < s \\ f(s) + (λ − s)f′(s) + g(λ − s), & s < λ ≤ s + t \end{matrix}

we obtain a path space structure on P = {(t, f) : t > 0, f ∈ P (t)} by way of π(t, f) = t, (s, f) · (t, g) = (s + t, f ∗ g).

Example 1.7. As we will see in section 4, the most important examples of path spaces arise as follows.

Let C be a separable Hilbert space, and consider the Hilbert space L^{2}((0, ∞); C) of all square integrable vector valued functions f : (0, ∞) → C with inner product

⟨f, g⟩ = ∫_{0}^{∞} ⟨f(x), g(x)⟩ dx.

For every t > 0, let P_{C}(t) denote the subspace of L^{2}((0, ∞); C) consisting of all functions f satisfying f(x) = 0 almost everywhere for x ≥ t. For f ∈ P_{C}(s), g ∈ P_{C}(t) we define the concatenation f ⊞ g ∈ P_{C}(s + t) by

f ⊞ g(λ) = \{ \begin{matrix}f(λ), & 0 < λ < s \\ g(λ − s), & s ≤ λ ≤ s + t. \end{matrix}

Let π : P_{C} → (0, ∞) be the total space defined by

(1.8)  P_{C} = {(t, f) : t > 0, f ∈ P_{C}(t)},

where π(t, f) = t. With multiplication (s, f) · (t, g) = (s + t, f ⊞ g), π : P_{C} → (0, ∞) becomes a path space.

Needless to say, there are obvious variations of these examples which can be formulated in different function spaces.

Remarks. The preceding examples involve generalized paths in vector spaces. It is less obvious how one might concatenate paths in multiply connected spaces so as to obtain a path space structure. For example, suppose we are given a closed set K ⊆ ℝ^{2} which represents an “obstruction” and one defines P (t) to be the set of all continuous functions

f : [0, t] → ℝ^{2} \ K

satisfying f(0) = 0 (we assume, of course, that 0 ∉ K). One may not concatenate elements according to the rule of example 1.5 since that rule of concatenation will not necessarily provide a path f ∗ g that avoids K, even if f and g both avoid K. Nevertheless, there are many ways to define such concatenation operations. We offer the following digression for the interested reader, though we will make no further reference to this class of examples in this paper.

Example 1.9. Let K be a closed subset of ℝ^{n}, n ≥ 2, having smooth boundary and which does not contain 0. Let

V : ℝ^{n} \ K → ℝ

be any C^{1} function which tends to +∞ near the boundary of K in the sense that if x_{n} is any sequence for which the distance from x_{n} to K tends to 0, then V (x_{n}) → +∞. For example, one may take

V (x) = sup_{y∈K} \frac{1}{|x − y|}, x ∈ ℝ^{n} \ K.

For such a V, the vector field

F (x) = −∇V (x)

becomes infinitely repulsive as x approaches K. If the vector v = ∇V (0) is not zero then we can replace V with

Ṽ (x) = V (x) − ∑_{k=1}^{n} v_{k}x_{k}

in order to achieve the normalization ∇V (0) = 0.

Let P (t) be the set of all continuously differentiable functions

f : [0, t] → ℝ^{n} \ K

satisfying f(0) = f′(0) = 0. We can use the vector field F to construct a concatenation operation on the family P = {(t, f) : t > 0, f ∈ P (t)} as follows. Starting with any continuous function ϕ : [0, t] → ℝ^{n} with ϕ(0) = 0, there is a corresponding element f ∈ P (t) defined as the solution of the initial value problem

f′(s) = ϕ(s) − ∇V (f(s)),
f(0) = 0.

Notice that f′(0) = 0 because of the normalization of V. There is no problem with the existence of global solutions of this differential equation because of the nature of V. Indeed, it is quite easy to show that this differential equation has a unique solution over the interval 0 ≤ s < ∞, and we obtain f by restricting the solution to the interval [0, t]. Conversely, every f ∈ P (t) arises in this way from the “driving” function ϕ(s) = f′(s) + ∇V (f(s)), 0 ≤ s ≤ t. Thus f ↔ ϕ defines a bijection of families of sets over (0, ∞).

Thus we can transfer the concatenation of example 1.5 directly to define a concatenation f, g ∈ P (s) × P (t) ↦ f ∗ g ∈ P (s + t). Explicitly, given f ∈ P (s), g ∈ P (t), then h = f ∗ g is the element of P (s + t) defined by the initial value problem

h′(λ) =ϕ(λ) − ∇V (h(λ))
h(0) =0

on the interval 0 ≤ λ ≤ s + t, where ϕ : [0, s + t] → ℝ^{n} is the driving function

ϕ(λ) = \{ \begin{matrix}f′(λ), & 0 ≤ λ < s \\ f′(s) + g′(λ − s) + ∇V (f(s)) + ∇V (g(λ − s)), & s ≤ λ ≤ s + t. \end{matrix}

Notice that f ∗ g agrees with f on the interval 0 ≤ λ ≤ s, but fails to agree with f(s) + g(λ − s) on the interval s ≤ λ ≤ s + t.

Remark. There are many potentials that one could use, and thus there are infinitely many path space structures whose fiber spaces are of the form

P (t) = {f : [0, t] → ℝ^{n} \ K : f ∈ C^{1}, f(0) = f′(0) = 0},

t > 0. It is clear from the construction that all of these path spaces are isomorphic to the example 1.5. On the other hand, notice that if one is presented with just the path space structure of one of these examples, it will not be possible to write down the relevant isomorphism if one does not know the correct potential V (at least up to a constant).

2. Additive forms and multiplicative forms.

Given a set X and a positive definite function B : X ×X → ℂ, one can construct a Hilbert space H(X, B). If B and C are two positive definite functions

B :X × X → ℂ
C :Y × Y → ℂ

then the product B ×C defines a positive definite function on the cartesian product (X × Y) × (X × Y) by way of

B × C((x_{1}, y_{1}), (x_{2}, y_{2})) = B(x_{1}, x_{2})C(y_{1}, y_{2}),

and one has a natural isomorphism of Hilbert spaces

H(X × Y, B × C) = H(X, B) ⊗ H(Y, C).

Thus, if one is given a path space π : P → (0, ∞) and a family of positive definite functions

B_{t} : P (t) × P (t) → ℂ, t > 0

which is multiplicative in an appropriate sense, then one would expect to obtain a continuous tensor product of Hilbert spaces...more precisely, a product system in the sense of [2].

The purpose of this section is to formulate these issues in the case where B_{t} is infinitely divisible in the sense that there is a family of conditionally positive definite functions

g_{t} : P (t) × P (t) → ℂ t > 0

such that B_{t} = e^{g_{t}}. Of course, multiplicative properties of B follow from appropriate additive properties of g. A classification of all product systems that can arise from this construction appears in Corollary 4.33.

We begin by reviewing a few basic facts and some terminology relating to conditionally positive definite functions [18]. Let X be a set and let g : X × X → ℂ be a function. g is called conditionally positive definite if it is self-adjoint in the sense that g(y, x) = \overline{g(x, y)} for all x, y, and is such that for every x_{1}, x_{2}, …, x_{n} ∈ X and λ_{1}, λ_{2}, …, λ_{n} ∈ ℂ with λ_{1} + λ_{2} + ⋯ + λ_{n} = 0, we have

∑_{i,j=1}^{n} λ_{i}\overline{λ}_{j}g(x_{i}, x_{j}) ≥ 0.

If p : X × X → ℂ is a positive definite function and ψ : X → ℂ is arbitrary then

(2.1)  g(x, y) = p(x, y) + ψ(x) + \overline{ψ(y)}

is conditionally positive definite. For g and p fixed, the function ψ satisfying (2.1) is not unique, but it is unique up to a perturbation of the form

ψ′(x) = ψ(x) + ic

where c is a real constant. Conversely, every conditionally positive definite function g can be decomposed into a sum of the form (2.1). For fixed g, the positive definite function appearing in (2.1) is not unique. If g : X ×X → ℂ is conditionally positive definite then

B(x, y) = e^{g(x,y)}

defines a positive definite function B : X × X → ℂ. The converse is false: there are self-adjoint functions g : X × X → ℂ which are not conditionally positive definite whose exponentials B = e^{g} are positive definite. However, if there is a sequence λ_{n} of positive numbers which tends to zero as n → ∞ and

(2.2)  B_{n}(x, y) = e^{λ_{n}g(x,y)}

is positive definite for every n = 1, 2, …, then g is conditionally positive definite. Notice that if

B_{n}(x, y) = e^{\frac{1}{n} g(x,y)}

is positive definite for every n = 1, 2, … then

B_{1}(x, y) = e^{g(x,y)} = B_{n}(x, y)^{n}

has a positive definite n^{th} root for every n = 1, 2, …; such a positive definite function B_{1} is called infinitely divisible.

Let P be a path space, which will be fixed throughout this section. P^{2} will denote the fiber product

P^{2} = {(t, x, y) : x, y ∈ P (t)},

with projection π(t, x, y) = t and fiber spaces P^{2}(t) = P (t) × P (t). A function g : P^{2} → ℂ is called conditionally positive definite if for every t > 0, the restriction of g to P (t) × P (t) is conditionally positive definite.

Given such a function g : P^{2} → ℂ, we may construct a Hilbert space H(t) for every t > 0. Briefly, letting ℂ_{0}P (t) denote the set of all functions f : P (t) → ℂ such that f(x) = 0 for all but finitely many x ∈ P (t) and which satisfy ∑_{x} f(x) = 0, then ℂ_{0}P (t) is a complex vector space and we may define a positive semidefinite sesquilinear form ⟨·, ·⟩ on ℂ_{0}P (t) by

⟨f_{1}, f_{2}⟩ = ∑_{x,y∈P (t)} f_{1}(x)\overline{f_{2}(y)}g(x, y).

After passing to the quotient of ℂ_{0}P (t) by the subspace {f ∈ ℂ_{0}P (t) : ⟨f, f⟩ = 0} and completing the resulting inner product space, we obtain a Hilbert space H(t). We will say that g is separable if H(t) is a separable Hilbert space for every t > 0.

In spite of the fact that each fiber P (t) is a lifeless set without additional structure, there is a useful notion of measurability for conditionally positive definite functions g : P^{2} → ℂ. Fix 0 < s < t. For every element y ∈ P (t) we may consider its associated propagator {y(λ, μ) : 0 ≤ λ < μ ≤ t}. Notice that for every x ∈ P (s) and every λ in the interval [0, t−s] we can form the complex number g(x, y(λ, λ+s).

Definition 2.2. g is called measurable if for every 0 < s < t < ∞, every pair of elements x_{1}, x_{2} ∈ P (s) and every y ∈ P (t),

λ ∈ (0, t − s) ↦ g(x_{1}, y(λ, λ + s)) − g(x_{2}, y(λ, λ + s))

defines a complex-valued Borel function.

Indeed, we will see that for the conditionally positive definite functions that are of primary interest, the functions appearing in Definition 2.2 are actually continuous (see Theorem 4.3). In view of the fact that we have imposed no structure on path spaces beyond that which follows from their rule of multiplication, this property appears noteworthy.

Finally, we introduce an appropriate notion of additivity for conditionally positive definite functions g : P^{2} → ℂ.

Definition 2.3. g is called additive if, there is a function defined on the full cartesian product

ψ : P × P → ℂ

such that for all s, t > 0, x_{1}, x_{2} ∈ P (s), y_{1}, y_{2} ∈ P (t),

(2.4)  g(x_{1}y_{1}, x_{2}y_{2}) − g(x_{1}, x_{2}) − g(y_{1}, y_{2}) = ψ(x_{1}, y_{1}) + \overline{ψ(x_{2}, y_{2})}.

Remarks. Notice that the domain of ψ, namely P × P, is larger than the domain of g, namely P^{2}. In fact, P^{2} is the diagonal of P × P:

P^{2} = {(p, q) ∈ P × P : π(p) = π(q)},

π : P → (o, ∞) being the natural projection.

The function ψ of Definition 2.3 is called the defect of g. The defect of g is not uniquely determined by equation (2.4), but if ψ_{1} and ψ_{2} both satisfy (2.4) then it is easy to see that there must be a function c : (0, ∞) × (0, ∞) → ℝ such that ψ_{2}(x, y) = ψ_{1}(x, y) + ic(s, t) for every x ∈ P (s), y ∈ P (t), s, t > 0. g is called exact if there is a function ρ : P → ℂ and a real-valued function c : (0, ∞) × (0, ∞) → ℝ such that for every s, t > 0, x ∈ P (s), y ∈ P (t),

(2.5)  ψ(x, y) = ρ(xy) − ρ(x) − ρ(y) + ic(s, t).

Notice that when g is exact we can replace it with

g_{0}(x, y) = g(x, y) − ρ(x) − \overline{ρ(y)}

to obtain a new conditionally positive definite function g_{0} : P^{2} → ℂ which obeys the addition formula (2.4) with zero defect.

Definition 2.6. Let P be a path space. An additive form is a function g : P^{2} → ℂ which restricts to a conditionally positive definite function on P (t) × P (t) for every t > 0, and which is separable, measurable, and additive.

A metric path space is a pair (P, g) consisting of a path space P and an additive form g : P^{2} → ℂ.

There are many natural examples of additive forms on path spaces. We give two simple ones here that are important for probability theory, and a third example which will be central to what follows.

For every t > 0 let P C[0, t] denote the space of all piecewise continuous realvalued functions f : [0, t] → ℝ and let P C be the path space

P C = {(t, f) : t > 0, f ∈ P C[0, t] }
π(t, f) = t

with concatenation defined by (s, f)(t, g) = (s + t, f ∗ g) where

f ∗ g(λ) = \{ \begin{matrix}f(λ), & 0 ≤ λ < s \\ g(λ − s), & s ≤ λ ≤ s + t. \end{matrix}

Let c be a positive constant.

Example 2.7: Gaussian forms on P C. For x_{1}, x_{2} ∈ P C[0, t], put

g(x_{1}, x_{2}) = −c ∫_{0}^{t}[x_{1}(λ) − x_{2}(λ)]^{2} dλ.

Example 2.8: Poisson forms on P C. Let h be a second positive constant and for x_{1}, x_{2} ∈ P C[0, t] put

g(x_{1}, x_{2}) = c ∫_{0}^{t}[e^{ih(x_{1}(λ)−x_{2}(λ))} − 1] dλ.

The forms g defined in examples 2.7 and 2.8 are essentially the covariance functions associated with random processes of the type indicated by their name. Note that in both cases the processes have stationary increments, and in fact 2.7 is the covariance function of Brownian motion. Notice too that the detailed structure of the path space used in these examples is not critical. For instance, if one replaces P C[0, t] with the corresponding Skorohod space D[0, t] and imitates what was done above, the new examples will share the essential features as those of 2.7 and 2.8.

More generally, with any continuous conditionally positive definite function of two real variables γ : ℝ^{2} → ℂ we can associate an additive form g on P C by way of

g(x_{1}, x_{2}) = ∫_{0}^{t} γ(x_{1}(λ), x_{2}(λ)) dλ,

for x_{1}, x_{2} ∈ P C[0, t].

Notice too that in all of the preceding examples the defect of g is zero. This will not be the case for additive forms that are associated with decomposable product systems as in in chapter II, and we will have to deal with additive forms having nonzero defects.

2.9: The standard examples. Let C be a separable Hilbert space and consider the path space P_{C} of example 1.7. In this case the additive form g : P_{C}^{2} → ℂ is simply the inner product inherited from L^{2}((0, ∞); C),

g(f_{1}, f_{2}) = ∫_{0}^{t} ⟨f_{1}(λ), f_{2}(λ)⟩ dλ,

for f_{1}, f_{2} ∈ P_{C}(t).

If we replace C with a Hilbert space C′ having the same dimension n as C then we obtain a new path space P_{C′} and a new form

g′ : P_{C′}^{2} → ℂ.

However, any unitary operator W : C → C′ induces an obvious isomorphism of path space structures

W̃ : P_{C} → P_{C′}

by way of W̃ f(λ) = W f(λ) for λ ∈ (0, t], t > 0. It follows that

g′(W̃ f_{1}, W̃ f_{2}) = g(f_{1}, f_{2})

for f_{1}, f_{2} ∈ P_{C}(t), t > 0. We conclude that, up to isomorphism, the examples of 2.9 depend only on the dimension n. This sequence of metric path spaces will be denoted (P_{n}, g_{n}), n = 1, 2, …, ∞. It is convenient to include the degenerate pair (P_{0}, g_{0}) where P_{0} is the trivial path space P_{0} = (0, ∞) × {0} with multiplication (s, 0)(t, 0) = (s + t, 0) and additive form g_{0}(x, y) = 0 for all x, y. Thus (P_{n}, g_{n}) is defined for every n = 0, 1, 2, …, ∞.

Suppose now that we are given a metric path space (P, g), and assume for the moment that g has defect zero. Then for every t > 0 we can define a positive definite function on P (t) by

x, y ∈ P (t) ↦ e^{g(x,y)}.

Let E(t) be the Hilvert space obtained from this positive definite function. In more detail, there is a function F_{t} : P (t) → E(t) with the property that E(t) is spanned by the range of F_{t} and

⟨F_{t}(x), F_{t}(y)⟩ = e^{g(x,y)}

for every x, y ∈ P (t). It can be seen that the separability hypothesis on g implies that E(t) is separable (in fact, the separability hypothesis implies that E(t) can be identified with a subspace of the symmetric Fock space over a separable one-particle space as at the end of section 4).

Let us examine the consequences of the formula (2.4) with ψ = 0. Fixing x_{1}, x_{2} ∈ P (s) and y_{1}, y_{2} ∈ P (t), we have

e^{g(x_{1}y_{1},x_{2}y_{2})} = e^{g(x_{1},x_{2})}e^{g(y_{1},y_{2})}

and hence

⟨F_{s+t}(x_{1}y_{1}), F_{s+t}(x_{2}y_{2})⟩ = ⟨F_{s}(x_{1}), F_{s}(x_{2})⟩ ⟨F_{t}(y_{1}), F_{t}(y_{2})⟩ .

It follows that there is a unique bilinear map (ξ, η) ∈ E(s) × E(t) ↦ ξ · η ∈ E(s + t) which satisfies F_{s}(x)·F_{t}(y) = F_{s+t}(xy) for all x ∈ P (s), y ∈ P (t) and this extended mapping acts like tensoring in that

⟨ξ_{1} · η_{1}, ξ_{2} · η_{2}⟩ = ⟨ξ_{1}, ξ_{2}⟩ ⟨η_{1}, η_{2}⟩

for every ξ_{1}, ξ_{2} ∈ E(s), η_{1}, η_{2} ∈ E(t).

Thus we can define an associative operation in the total family of Hilbert spaces p : E → (0, ∞) defined by

E = {(t, ξ) : t > 0, ξ ∈ E(t) }
p(t, ξ) = t

by way of (s, ξ)(t, η) = (s + t, ξ · η). This structure p : E → (0, ∞) has the main features of a product system [2]. However, since the total family E carries no natural Borel structure (because we are not given a Borel structure on the total space of P), the measurability axioms for product systems are meaningless here. For this reason, we will refer to p : E → (0, ∞) as the product structure associated with the metric path space (P, g).

The above construction required that g have defect zero. If g has nonzero defect but is exact, then one can modify this construction so as to obtain a product structure in this case as well (see section 4).

In general, Theorem 4.3 below implies that every additive form on a path space is exact. Moreover, we will find that (P, g) must be “essentially” isomorphic to one of the standard examples (P_{n}, g_{n}), n = 0, 1, 2, …, ∞ in such a way that the product structure associated with (P, g) is either the trivial one having one-dimensional spaces E(t) or is isomorphic to one of the standard product systems of [2]. Thus, every metric path space gives rise to a product structure that is completely understood.

3. Exactness of cocycles. Let C be a separable Hilbert space and let L be the space of all measurable functions f : (0, ∞) → C which are locally square integrable in the sense that

∫_{0}^{T} ∥f(x)∥^{2} dx < ∞

for every T > 0. The topology of L is defined by the sequence of seminorms

∥f∥_{n} = (∫_{0}^{n} ∥f(x)∥^{2} dx)^{1/2}

n = 1, 2, … and

d(f, g) = ∑_{1}^{∞} 2^{−n} \frac{∥f − g∥_{n}}{1 + ∥f − g∥_{n}}

is a translation invariant metric on L with respect to which it becomes a separable Frèchet space.

L is the dual of the inductive limit of Hilbert spaces

L_{0} = ⋃_{T >0} L^{2}((0, t); C).

L_{0} is identified with the submanifold of L^{2}((0, ∞); C) consisting of all functions having compact support; a net f_{α} ∈ L_{0} converges to f ∈ L_{0} iff there is a T > 0 such that f_{α} is supported in (0, T) for sufficiently large α, and

lim_{α} ∫_{0}^{T} ∥f_{α}(x) − f(x)∥^{2} dx = 0.

L is isometrically anti-isomorphic to the dual of L_{0} by way of the sesquilinear pairing

f, g ∈ L_{0} × L ↦ ⟨f, g⟩ = ∫_{0}^{∞} ⟨f(x), g(x)⟩ dx.

A function t ∈ (0, ∞) ↦ ϕ_{t} ∈ L is called measurable if it is a Borel function relative to the weak^{∗} topology on L, i.e.,

t ∈ (0, ∞) ↦ ⟨f, ϕ_{t}⟩

should be a complex-valued Borel function for every f ∈ L_{0}.

Definition 3.1. An additive cocycle is a measurable function t ∈ (0, ∞) ↦ ϕ_{t} ∈ L satisfying

ϕ_{s+t}(x) = ϕ_{s}(x) + ϕ_{t}(x + s) a.e.(dx)

for every s, t > 0.

The purpose of this section is to prove the following characterization of additive cocycles.

Theorem 3.2. Let {ϕ_{t} : t > 0} be an additive cocycle. Then there is a function f ∈ L such that

ϕ_{t}(x) = f(x + t) − f(x) a.e.(dx)

for every t > 0.

Remarks. Theorem 3.2 bears a resemblance to known results about multiplicative cocycles associated with transitive actions of topological groups on topological spaces [20]. However, our setting here differs in several key aspects. Rather than a group action we have a transitive action of the additive semigroup of positive reals. Moreover, the elements of L must satisfy nontrivial integrability conditions.

Notice too that we make no assumption about the boundedness of {ϕ_{t} : t > 0}. That is to say, we do not assume that

sup_{t>0} | ⟨f, ϕ_{t}⟩ | ≤ M_{f} < ∞

for every f ∈ L_{0}. Indeed, it is easy to give examples of cocycles for which this condition is not satisfied. The fact that additive cocycles can be unbounded means that softer techniques that require some form of boundedness (specifically, techniques involving the use of Banach limits on the additive semigroup ℝ^{+}) are not available. For these reasons we have taken some care to give a complete proof of Theorem 3.2.

Remark on measurability. Every weak^{∗} measurable function t ↦ ϕ_{t} ∈ L is a Borel mapping of metric spaces. This means that for every g ∈ L, the function

t ∈ (0, ∞) ↦ d(ϕ_{t}, g)

is Borel measurable. To see that, notice that for each n = 1, 2, …,

∥ϕ_{t} − g∥_{n} = sup_{k≥1} | ⟨f_{k}, ϕ_{t} − g⟩ |,

the supremum extended over a countable set of functions f_{1}, f_{2}, … which is dense in the unit ball of L^{2}((0, n); C). Hence t ↦ ∥ϕ_{t} − g∥_{n} is a Borel function for each n. It follows that

d(ϕ_{t}, g) = ∑_{n=1}^{∞} 2^{−n} \frac{∥ϕ_{t} − g∥_{n}}{1 + ∥ϕ_{t} − g∥_{n}}

is a Borel function of t.

The following lemma asserts that additive cocycles are metrically continuous. I am indebted to Calvin Moore for pointing out a pertinent reference [12, Théorème 4, p. 23] for a key step in its proof.

Lemma 3.3. Let {ϕ_{t} : t > o} be an additive cocycle. Then t ↦ ϕ_{t} moves continuously in the metric topology of L and we have

lim_{t→0+} ϕ_{t} = 0.

proof. Let {T_{t} : t ≥ 0} be the natural translation semigroup which acts in L by

T_{t}f(x) = f(x + t).

For every n = 1, 2, … and every f ∈ L we have

lim_{t→0} ∥T_{t}f − f∥_{n}^{2} = lim_{t→0} ∫_{0}^{n} ∥f(x + t) − f(x)∥^{2} dx = 0,

and hence d(T_{t}f, f) → 0 as t → 0. Thus the semigroup {T_{t} : t ≥ 0} is continuous in the metric topology of L.

By the preceding remark, the function

t ∈ (0, ∞) ↦ ϕ_{t} ∈ L

is a Borel function taking values in a separable complete metric space. Thus there is a subset N ⊆ (0, ∞) of the first category in (0, ∞) such that the restriction of this function to (0, ∞) \ N is metrically continuous [19, p. 306].

Let t_{0} ∈ (0, ∞) and choose t_{n} ∈ (0, ∞) such that t_{n} → t_{0} as n → ∞. Since

M = ∪_{n=0}^{∞}N − t_{n}

is a first category subset of (0, ∞) we may find s ∈ (0, ∞) \ M. Then s + t_{n} belongs to (0, ∞) \ N for every n = 0, 1, 2, …, and it follows that

lim_{n→∞} d(ϕ_{s+t_{n}}, ϕ_{s+t_{0}}) = 0.

Writing

ϕ_{t_{0}} + T_{t_{0}}ϕ_{s} = ϕ_{t_{0}+s} = lim_{n→∞} ϕ_{t_{n}+s} = lim_{n→∞}(ϕ_{t_{n}} + T_{t_{n}}ϕ_{s})

and noting that by the metric continuity of {T_{t}} we have

lim_{n→∞} T_{t_{n}}ϕ_{s} = T_{t_{0}}ϕ_{s},

we conclude that

ϕ_{t_{0}} = lim_{n→∞} ϕ_{t_{n}}.

For continuity at t = 0+, notice that for every s, t_{0} > 0 we have

ϕ_{s+t_{0}} = ϕ_{s} + T_{s}ϕ_{t_{0}}.

Letting s → 0+ and noting that ϕ_{s+t_{0}} → ϕ_{s+t_{0}} and T_{s}ϕ_{t_{0}} → ϕ_{t_{0}}, we obtain

lim_{s→0+} ϕ_{s} = 0,

as required □

proof of Theorem 3.2. By the Fubini theorem we may find a Borel function

v : (t, x) ∈ (0, ∞) × (0, ∞) ↦ v(t, x) ∈ C

such that v(t, x) = ϕ_{t}(x) almost everywhere with respect to the product measure dt dx on (0, ∞) × (0, ∞). Because of the cocycle condition

ϕ_{s+t}(x) = ϕ_{s}(x) + ϕ_{t}(x + s) a.e. (dx)

it follows that v must satisfy

v(s + t, x) = v(s, x) + v(t, x + s)

for almost every triple (s, t, x) ∈ (0, ∞)^{3} with respect to ds dt dx. By the Fubini theorem there is a Borel set N ⊆ (0, ∞) of measure zero such that for every x ∈ (0, ∞) \ N we have

v(s + t, x) = v(s, x) + v(t, x + s)

almost everywhere ds dt. Choose a sequence x_{1} > x_{2} > … in (0, ∞) \ N which decreases to 0.

We claim that there is a sequence of Borel functions f_{n} : (x_{n}, ∞) → C satisfying

(3.5.1)  f_{n}(ξ + t) − f_{n}(ξ) = v(t, ξ)

almost everywhere on (x_{n}, ∞) × (0, ∞) with respect to dξ dt, and which is coherent in that

(3.5.2)  f_{n+1} ↾_{(x_{n},∞)}= f_{n}  almost everywhere.

Proceeding inductively, we define f_{1} by

f_{1}(ξ) = v(ξ − x_{1}, x_{1}), ξ > x_{1}.

To verify (3.5.1), note that for λ, t > 0 and ξ = λ + x_{1} we have

f_{1}(λ + x_{1} + t) − f_{1}(λ + x_{1}) = v(λ + t, x_{1}) − v(λ, x_{1})

which by (3.4) is almost everywhere (dλ dt) equal to v(t, x_{1} + λ), hence we have (3.5.1).

Assuming now that f_{k} has been defined for 1 ≤ k ≤ n, define g : (x_{n+1}, ∞) → C by

g(ξ) = v(ξ − x_{n+1}, x_{n+1}).

If we replace x_{1} with x_{n+1} in the argument of the preceding paragraph we obtain the conclusion

g(ξ + t) − g(ξ) = v(t, ξ)

almost everywhere (dξ dt) on (x_{n+1}, ∞) × (0, ∞). Restricting ξ to the interval (x_{n}, ∞) ⊆ (x_{n+1}, ∞) then gives

g(ξ + t) − g(ξ) = v(t, ξ) = f_{n}(ξ + t) − f_{n}(ξ)

almost everywhere (dξ dt) on (x_{n}, ∞) × (0, ∞). Thus the function h : (x_{n}, ∞) → C defined by h(x) = g(x) − f_{n}(x) is Borel-measurable and translation invariant in the sense that for almost every t ≥ 0 we have

h(ξ + t) = h(ξ) a.e. (dξ).

It follows that there is a vector c ∈ C such that

h(ξ) = c a.e. (dξ).

If we set f_{n+1} = g(x) − c then we obtain both required conditions (3.5.1) and (3.5.2).

Because of the coherence property (3.5.2), there is a Borel function f : (0, ∞) → C satisfying

f(ξ) = f_{n}(ξ) a.e. (dξ)

on (x_{n}, ∞), for every n = 1, 2, …; and because of (3.5.1) we have

f(ξ + t) − f(ξ) = v(t, ξ)

for almost every pair (ξ, t) ∈ (0, ∞) × (0, ∞)) with respect to the product measure dξ dx.

It follows that

f(ξ + t) − f(ξ) = ϕ_{t}(ξ)

almost everywhere dξ dt, and by another application of Fubini’s theorem we may conclude that there is a Borel set N ⊆ (0, ∞) such that for all t ∈ (0, ∞) \ N we have

(3.6)  ϕ_{t}(x) = f(x + t) − f(x)  a.e. (dx).

We show next that the exceptional set N of (3.6) can be eliminated. To that end, consider the vector space F consisting of all Borel functions

F : (0, ∞) → C,

where we make the traditional identification of two functions that agree almost everywhere. We endow this space with the topology of local convergence in measure. More precisely, a net f_{α} ∈ F converges to F ∈ F iff for every T > 0, the restrictions F_{α} ↾_{(0,T)} converge in Lebesgue measure to F ↾_{(0,T)}. F is metrizable as a separable complete metric space by way of

d(F, G) = ∑_{n=1}^{∞} 2^{−n}d_{n}(F, G)

where for n = 1, 2, …,

d_{n}(F, G) = ∫_{0}^{n} \frac{∥F (x) − G(x)∥}{1 + ∥F (x) − G(x)∥} dx.

The translation semigroup {T_{t} : t ≥ 0} defined by

T_{t}F (x) = F (x + t)

acts continuously on the Frèchet space F.

It follows from these remarks that the right side of (3.6) is continuous in t, provided that we consider T_{t}f − f as an element of F. Since the inclusion map L ⊆ F is continuous, it follows from Lemma 3.3 that the left side of (3.6) defines a continuous function

t ∈ (0, ∞) ↦ ϕ_{t} ∈ F.

Equation (3.6) implies that these two continuous functions agree on the complement of a null set, and hence they agree for all t > 0.

Thus, we have a Borel function f : (0, ∞) → L with the property that for every t > 0,

(3.7)  ϕ_{t}(x) = f(x + t) − f(x)  a.e. (dx).

It remains to show that the condition ϕ_{t} ∈ L for every t > 0 implies that f itself belongs to L. We will deduce that from the following result, the proof of which is based on an argument shown to me by Henry Helson, who kindly consented to its inclusion in this paper.

Lemma 3.8. Let f be a nonnegative Borel function defined on (0, ∞) satisfying

∫_{0}^{T} |f(x + t) − f(x)|^{2} dx < ∞

for every T > 0 and every t > 0. Then for every T > 0 we have

∫_{0}^{T} |f(x)|^{2} dx < ∞.

proof. Fix T > 0. It suffices to show that for every nonnegative funtion g ∈ L^{2}(0, T) we have

∫_{0}^{T} f(x)g(x) dx < ∞.

To that end, find a function u ∈ L^{2}(0, 1) such that u(x) > 0 for every x and

∫_{0}^{1} f(x)u(x) dx < ∞,

and define a function Φ : (0, ∞) → ℝ^{+} by

Φ(t) = ∫_{0}^{1} |f(x + t) − f(x)|u(x) dx.

We claim that Φ is continuous and tends to 0 as t → 0+. Indeed, we may apply Lemma 3.3 to the additive cocycle

ϕ_{t}(x) = f(x + t) − f(x)

in the case where C is the complex numbers to conclude that t ∈ (0, ∞) ↦ ϕ_{t} ∈ L is metrically continuous and tends to 0 as t → 0+. Since F ∈ L ↦ |F | ∈ L is clearly continuous, the same is true of the modulus t ∈ (0, ∞) ↦ |ϕ_{t}| ∈ L, and the claim follows.

Notice next that for every positive g ∈ L^{2},

(3.10)  ∫_{0}^{1} ∫_{0}^{T} f(x + t)g(t)u(x) dt dx < ∞.

Indeed, since

f(x + t)g(t)u(x) ≤ |ϕ_{t}(x)|g(t)u(x) + f(x)g(t)u(x),

and since

∫_{0}^{1} ∫_{0}^{T} |ϕ_{t}(x)|g(t)u(x) dt dx = ∫_{0}^{T} Φ(t)g(t) dt < ∞

and

∫_{0}^{1} ∫_{0}^{T} f(x)g(t)u(x) dt dx = ∫_{0}^{T} f(x)u(x) dx ∫_{0}^{1} g(t) dt < ∞,

(3.10) follows.

Since u > 0, (3.10) implies that

∫_{0}^{T} f(x + t)g(t) dt < ∞

for almost every x ∈ (0, 1). Nor for every x > 0 we have

∫_{0}^{T} f(t)g(t) dt ≤ ∫_{0}^{T} f(x + t)g(t) dt + ∫_{0}^{T} |f(x + t) − f(x)|g(t) dt.

The first integral on the right is finite for certain values of x ∈ (0, 1) and the second one is finite for all x > 0. Thus (3.9) follows. □

To complete the proof of theorem 3.2, we must show that any measurable function f : (0, ∞) → C for which the differences ϕ_{t}(x) = f(x + t) − f(x) belong to L must itself belong to L. Equivalently, if f : (0, ∞) → C is a Borel function for which

∫_{0}^{T} ∥f(x + t) − f(x)∥^{2} dx < ∞

for every T > 0 and every t > 0, then

(3.11)  ∫_{0}^{T} ∥f(x)∥^{2} dx < ∞

for every T > 0.

To see that, fix such an f and consider F (x) = ∥f(x)∥. We have

|F (x + t) − F (x)| ≤ ∥f(x + t) − f(x)∥,

hence F satisfies the hypotheses of Lemma 3.8. It follows from Lemma 3.8 that F is locally in L^{2}. □

4. Classification of additive forms. Before stating the main result on classification of metric path spaces we introduce the concept of a strongly spanning set. Let H be a Hilbert space and let S be a subset of H. We will write e^{H} for the symmetric Fock space over the one-particle space H,

e^{H} = \begin{matrix}∞ \\ ∑ & ⊕ \\ n=0 \end{matrix} H^{(n)},

H^{(n)} denoting the symmetric tensor product of n copies of H when n ≥ 1, and where H^{(0)} is defined as ℂ. Consider the exponential map exp : H → e^{H}, defined by

exp(ξ) = ∑_{n=0}^{∞} \frac{1}{\sqrt{n!}} ξ^{n},

ξ^{n} denoting ξ^{⊗n} if n ≥ 1 and ξ^{0} = 1 ∈ ℂ. It is well known that e^{H} is spanned by

exp(H) = {exp(ξ) : ξ ∈ H}.

Definition 4.1. A subset S ⊆ H is said to strongly span H if e^{H} is spanned by the set of vectors exp(S) = {exp(ξ) : ξ ∈ S}.

Remarks. When we use the term ‘span’, we of course mean closed linear span. Every strongly spanning set must span H, but the converse if false, as the following remarks show. In general, every vector ζ ∈ e^{H} give rise to a holomorphic function f_{ζ} : H → ℂ by way of

f_{ζ}(ξ) = ⟨exp(ξ), ζ⟩ .

Indeed, if we let ζ_{n} be the projection of ζ onto H^{(n)}, then we have a representation of f_{ζ} as a power series

f_{ζ}(ξ) = ∑_{0}^{∞} \frac{1}{\sqrt{n!}} ⟨ξ^{n}, ζ_{n}⟩ ,

and because ∑_{n} ∥ζ_{n}∥^{2} = ∥ζ∥^{2} < ∞, this power series converges absolutely and uniformly over the ball in H of radius R, for every R > 0. Thus, F = {f_{ζ} : ζ ∈ e^{H}} is a complex vector space of entire functions defined on H. S is a strongly spanning set iff

F ∈ F, F (S) = {0} ⟹ F = 0.

If H is finite dimensional then every holomorphic polynomial F : H → ℂ belongs to F. In particular, if H is two dimensional and {e_{1}, e_{2}} is an orthonormal basis for H then

S = {λ(e_{1} + e_{2}) : λ ∈ ℂ} ∪ {λ(e_{1} − e_{2}) : λ ∈ ℂ}

clearly spans H because it contains e_{1} + e_{2} and e_{1} − e_{2}. On the other hand, S is not strongly spanning because

F (ξ) = ⟨ξ, e_{1}⟩^{2} − ⟨ξ, e_{2}⟩^{2}

is a nonzero polynomial which vanishes on S. More generally, any spanning subset S of a finite dimensional H which is contained in the zero set of a nontrivial polynomial will fail to be strongly spanning. This will be the case whenever S is an algebraic set, or an algebraic variety in H.

The following is our main classification of metric path spaces.

Theorem 4.3. Let P be a path space and let g : P^{2} → ℂ be a separable measurable additive form. Then there is a separable Hilbert space C, a complex-valued function ρ : P → ℂ and a mapping of fiber spaces

log : P → P_{C}

such that log(xy) = log(x) ⊞ log(y) for every x, y ∈ P, and such that for every t > 0, x_{1}, x_{2} ∈ P (t),

(4.3.1)  log(P (t)) strongly spans P_{C}(t)
(4.3.2)  g(x_{1}, x_{2}) = ⟨log(x_{1}), log(x_{2})⟩ + ρ(x_{1}) + \overline{ρ(x_{2})}

Remarks. The assertion that log is a mapping of fiber spaces means log(P (t)) ⊆ P_{C}(t) for every t > 0. Thus, log defines a homomorphism of the path space structure of P into that of P_{C}. Property 4.3.1 asserts that, even though log(P (t)) may not be dense in P_{C}(t), it is a rich enough subset so that

\overline{span} exp(P (t)) = e^{P_{C}(t)}.

Finally, notice that property 4.3.2 implies that g is an exact form in the sense of (2.5). Indeed, if we take x_{1}, x_{2} ∈ P (s) and y_{1}, y_{2} ∈ P (t) then by definition of the concatenation operation ⊞ in P_{C} we have

⟨log(x_{1}) ⊞ log(y_{1}), log(x_{2}) ⊞ log(y_{2})⟩ = ⟨log(x_{1}), log(x_{2}⟩ + ⟨log(y_{1}), log(y_{2}⟩ .

Using the fact that log(x_{k}y_{k}) = log(x_{k}) ⊞ log(y_{k}) for k = 1, 2 we find that

g(x_{1}y_{1}, x_{2}y_{2}) − g(x_{1}, x_{2}) − g(y_{1}, y_{2}) = ψ(x_{1}, y_{1}) + \overline{ψ(x_{2}, y_{2})}

where ψ(x, y) = ρ(xy) − ρ(x) − ρ(y) and ρ is the function given by 4.3.2.

The proof of Theorem 4.3 will occupy most of the remainder of this section, and will proceed along the following lines. We first use g to associate a Hilbert space H_{t} with P (t) for every t > 0. We then show that for s < t, H_{s} embeds naturally in H_{t} so that we can form an inductive limit of Hilbert spaces

H_{∞} = lim_{→} H_{t}.

We introduce a strongly continuous semigroup of isometries acting in H_{∞} which will turn out to be pure. This implies that H_{∞} can be coordinatized in such a way that it becomes an L^{2} space of vector valued functions

H_{∞} = L^{2}((0, ∞); C)

and the semigroup of isometries is the natural shift semigroup. Finally, we use the results of section 3 to solve a cohomological problem. Once that is accomplished we can define the required “logarithm” log : P → P_{C} and verify its properties.

Definition of H_{t}. Fix t > 0. Let ℂ_{0}P (t) denote the complex vector space of all finitely nonzero functions f : P (t) → ℂ satisfying the condition ∑_{x} f(x) = 0, and let ⟨·, ·⟩ be the sesquilinear form defined on ℂ_{0}P (t) by

⟨f, g⟩ = ∑_{x,y∈P (t)} f(x)\overline{g(y)}g(x, y).

⟨·, ·⟩ is positive semidefinite, and after passing to the quotient of ℂ_{0}P (t) by the subspace of null functions K_{t} = {f : ⟨f, f⟩ = 0} we obtain an inner product space, whose completion is denoted H_{t}.

Now ℂ_{0}P (t) is spanned by the set of all differences {δ_{x} − δ_{y} : x, y ∈ P (t)}, δ_{z} denoting the unit function

δ_{z}(u) = \{ \begin{matrix}1, & if u = z \\ 0, & otherwise. \end{matrix}

Hence H_{t} is spanned by the set {[x] − [y] : x, y ∈ P (t)}, where [x] − [y] denotes the element δ_{x} − δ_{y} + K_{t}. The inner product in H_{t} is characterized by

(4.4)  ⟨[x_{1}] − [y_{1}], [x_{2}] − [y_{2}]⟩ = g(x_{1}, x_{2}) − g(x_{1}, y_{2}) − g(y_{1}, x_{2}) + g(y_{1}, y_{2}),

for x_{1}, x_{2}, y_{1}, y_{2} ∈ P (t). Notice that although we have written [x] − [y] as if it were a difference, it is not actually the difference it appears to be since [x] and [y] do not belong to H_{t}. It is in fact a two-variable function which satisfies a certain cocycle identity. But the notation is convenient provided one is careful never to treat [x] and [y] as if they were elements of H_{t}.

H_{t} is separable because of our separability hypothesis on g (See the discussion preceding Definition 2.2). When it is necessary to distinguish between the various inner products we will write ⟨·, ·⟩_{t} for the inner product on H_{t}.

Embedding H_{s} in H_{t} for s < t. Fix s, t with 0 < s < t and choose an element e ∈ P (t − s). We want to show that there is an isometric linear map of H_{s} into H_{t} which carries differences of the form [x_{1}]−[x_{2}] with x_{i} ∈ P (s) into [x_{1}e]−[x_{2}e], and moreover that this isometry does not depend on the particular choice of e ∈ P (t−s). To that end, we claim that for all x_{i}, y_{i} ∈ P (s), and z_{i} ∈ P (t − s) i = 1, 2, we have

(4.5)  ⟨[x_{1}e] − [x_{2}e], [y_{1}z_{1}] − [y_{2}z_{2}]⟩_{t} = ⟨[x_{1}] − [x_{2}], [y_{1}] − [y_{2}]⟩_{s}

Indeed, because of (4.4) the left side is

(4.6)  g(x_{1}e, y_{1}z_{1}) − g(x_{1}e, y_{2}z_{2}) − g(x_{2}e, y_{1}z_{1}) + g(x_{2}e, y_{2}z_{2})

But by (2.4) we have,

g(x_{i}e, y_{j}z_{j}) = g(x_{i}, y_{j}) + g(e, z_{j}) + ψ(x_{i}, e) + \overline{ψ(y_{j}, z_{j})}.

It follows that for j = 1, 2,

g(x_{1}e, y_{j}z_{j}) − g(x_{2}e, y_{j}z_{j}) = g(x_{1}, y_{j}) − g(x_{2}, y_{j}) + ψ(x_{1}, e) − ψ(x_{2}, e).

When we subtract this expression for j = 2 from the expression for j = 1 the terms involving ψ cancel and we are left with

g(x_{1}, y_{1}) − g(x_{1}, y_{2}) − g(x_{2}, y_{1}) + g(x_{2}, y_{2})

which is the right side of (4.5).

By taking z_{1} = z_{2} = e in (4.5) we see that

⟨[x_{1}e] − [x_{2}e], [y_{1}e] − [y_{2}e]⟩_{t} = ⟨[x_{1}] − [x_{2}], [y_{1}] − [y_{2}]⟩_{s}

and hence there is a unique linear isometry V (t, s) : H_{s} → H_{t} such that

V (t, s)([x_{1}] − [x_{2}]) = [x_{1}e] − [x_{2}e]

for every x_{1}, x_{2} ∈ P (s). Moreover, since H_{t} is spanned by elements of the form [y_{1}z_{1}] − [y_{2}z_{2}] for y_{i} ∈ P (s), z_{i} ∈ P (t − s) (here we use the fact that P (t) = P (s)P (t−s)), it also follows from (4.5) that V (t, s) is independent of the particular choice of e ∈ P (t − s); in more concrete terms, if e and f are two elements of P (t − s), then for all x_{1}, x_{2} ∈ P (s) we have

[x_{1}e] − [x_{2}e] = [x_{1}f] − [x_{2}f].

It follows from the latter that we have the consistency relation

V (t, s)V (s, r) = V (t, r)

for all 0 < r < s < t. Indeed, if we choose e_{1} ∈ P (r − s) and e_{2} ∈ P (t − s) then for x_{1}, x_{2} ∈ P (r) we have

V (t, s)V (s, r)([x_{1}] − [x_{2}]) = V (t, s)([x_{1}e_{1}] − [x_{2}e_{1}]) = [x_{1}e_{1}e_{2}] − [x_{2}e_{1}e_{2}],

and the right side is V (t, r)([x_{1}]−[x_{2}]) simply because e_{1}e_{2} is an element of P (t−s).

Thus we can form the inductive limit of inner product spaces

lim_{→} H_{t}.

Explicitly, this consists of all functions t ∈ (0, ∞) ↦ ξ_{t} ∈ H_{t} having the property that there is a T = T_{ξ} > 0 such that for all t > T,

ξ_{t} = V (t, T)ξ_{T} .

The inner product in the inductive limit is defined by

⟨ξ, η⟩ = lim_{t→∞} ⟨ξ_{t}, η_{t}⟩_{t} .

H_{∞} is defined as the completion of

lim_{→} H_{t},

and it is a separable Hilbert space.

Choose x_{1}, x_{2} ∈ P (t). By a slight abuse of notation we will write [x_{1}] − [x_{2}] for the element of H_{∞} defined by the function

ξ_{λ} = \{ \begin{matrix}0, & for λ ≤ t \\ V (λ, t)([x_{1}] − [x_{2}]), & for λ > t. \end{matrix}

Notice that by definition of H_{∞}, we will have

[x_{1}e] − [x_{2}e] = [x_{1}] − [x_{2}]

for every x_{1}, x_{2} ∈ P (t), t > 0 and for an arbitrary element e of P. H_{∞} is spanned by the set of formal differences

{[x_{1}] − [x_{2}] : x_{i} ∈ P (t), t > 0}.

Finally, note that the inner product in H_{∞} is defined by its values on these formal differences as follows. Choose s ≠ t, x_{1}, x_{2} ∈ P (s) and y_{1}, y_{2} ∈ P (t). In order to evaluate the inner product ⟨[x_{1}] − [x_{2}], [y_{1}] − [y_{2}]⟩ we may suppose that s < t. Choose any e ∈ P (t − s). Then since [x_{1}] − [x_{2}] = [x_{1}e] − [x_{2}e] we have

⟨[x_{1}] − [x_{2}], [y_{1}] − [y_{2}]⟩ = ⟨[x_{1}e] − [x_{2}e], [y_{1}] − [y_{2}]⟩
= g(x_{1}e, y_{1}) − g(x_{1}e, y_{2}) − g(x_{2}e, y_{1}) + g(x_{2}e, y_{2}).

The subspaces N_{t} ⊆ H_{∞}. For every t > 0 we define a subspace N_{t} of H_{∞} as follows

N_{t} = \overline{span}\{[x_{1}] − [x_{2}] : x_{i} ∈ P (t)\}.

Choose 0 < s < t and x_{1}, x_{2} ∈ P (s). The preceding remarks imply that the element of H_{∞} represented by the difference [x_{1}] − [x_{2}] can be identified with a difference [y_{1}] − [y_{2}] of elements from P (t) by taking y_{i} = x_{i}e for some e ∈ P (t − s). It follows that the spaces N_{t} are increasing

(4.7.1.)  0 ≤ s ≤ t ⟹ N_{s} ⊆ N_{t}

Moreover, since the images of the spaces H_{t}, t > 0 in H_{∞} span H_{∞} we also have

(4.7.2)  \overline{∪_{t>0}N_{t}} = H_{∞}.

The semigroup {U_{t} : t ≥ 0}. We now introduce a semigroup of isometries {U_{t} : t ≥ 0} acting on H_{∞}. Fix t > 0 and choose an element e ∈ P (t).

There is a formula analogous to (4.5) in which the order of multiplication is reversed. That is, for 0 < s < t, x_{i}, y_{i} ∈ P (s) and e, z_{i} ∈ P (t − s) we claim that

(4.8)  ⟨[ex_{1}] − [ex_{2}], [z_{1}y_{1}] − [z_{2}y_{2}]⟩_{t} = ⟨[x_{1}] − [x_{2}], [y_{1}] − [y_{2}]⟩_{s}

Notice that the inner product on the left (resp. right) is taken in the Hilbert space H_{t} (resp. H_{s}). The proof of (4.8) is the same as the proof of (4.5). The identity (4.8) implies that if we choose an element f ∈ P (t) for some t > 0 then for any s > 0 and any pair of elements x_{1}, x_{2} ∈ P (s), the element of H_{∞} defined by [fx_{1}] − [fx_{2}] does not depend on the particular choice of f ∈ P (t) in that for every g ∈ P (t) we have

(4.9)  [fx_{1}] − [fx_{2}] = [gx_{1}] − [gx_{2}].

Note that in (4.9) the vectors on both sides belong to H_{∞}. Moreover, if y_{1}, y_{2} is a second pair of elements in P (s) then (4.8) also implies

⟨[fx_{1}] − [fx_{2}], [fy_{1}] − [fy_{2}]⟩ = ⟨[x_{1}] − [x_{2}], [y_{1}] − [y_{2}]⟩ .

Now fix t > 0 and choose f ∈ P (t). It follows that there is a unique isometry U_{t,s} : N_{s} → N_{s+t} which satisfies

U_{t,s}([x_{1}] − [x_{2}]) = [fx_{1}] − [fx_{2}].

Because of (4.9), U_{t,s} does not depend on the choice of f. Notice too that U_{t,s} does not depend on s. Indeed, if 0 < s_{1} < s_{2} and x_{1}, x_{2} ∈ P (s_{1}) then for any element z ∈ P (s_{2} − s_{1}) we have

U_{t,s_{1}}([x_{1}] − [x_{2}]) = [fx_{1}] − [fx_{2}] = [(fx_{1})z] − [(fx_{2})z],

while

U_{t,s_{2}}([x_{1}] − [x_{2}]) = U_{t,s_{2}}([x_{1}z] − [x_{2}z]) = [f(x_{1}z)] − [f(x_{2}z)].

The right hand sides of these two formulas agree because of the associativity of the multiplication in P.

By (4.7.2) there is a unique isometry U_{t} : H_{∞} → H_{∞} satisfying

U_{t}([x_{1}] − [x_{2}]) = [fx_{1}] − [fx_{2}]

for all x_{1}, x_{2} ∈ P (s), s > 0. For t = 0 we set U_{0} = 1. Finally, note that {U_{t} : t ≥ 0} is a semigroup. Indeed, given s, t > 0 we choose f ∈ P (s) and g ∈ P (t), and note that for every x_{1}, x_{2} ∈ P (r) we have

U_{s}U_{t}([x_{1}] − [x_{2}]) = U_{s}([gx_{1}] − [gx_{2}]) = [fgx_{1}] − [fgx_{2}].

The right side must be U_{s+t}([x_{1}]−[x_{2}]) because the product fg belongs to P (s+t). This shows that U_{s}U_{t} = U_{s+t} on each N_{r}, and by (4.7.2) it follows that U_{s}U_{t} = U_{s+t}.

Strong continuity. Since H_{∞} is a separable Hilbert space, the strong continuity of {U_{t} : t ≥ 0} will follow is we prove that the operator function

t ∈ (0, ∞) ↦ U_{t} ∈ B(H_{∞})

is weakly measurable in the sense that ⟨U_{λ}ξ, η⟩ defines a Borel function on 0 < λ < ∞ for every ξ, η ∈ H_{∞} [2, Proposition 2.5 (ii)]. In turn, because of the semigroup property it suffices to verify this for λ restricted to the interval 0 < λ ≤ 1. To that end, we claim that for any pair of vectors ξ_{1}, ξ_{2} in the spanning set

⋃_{t>0}{[x_{1}] − [x_{2}] : x_{i} ∈ P (t)},

the function

(4.10)  λ ∈ (0, 1] ↦ ⟨U_{λ}ξ_{1}, ξ_{2}⟩

is Borel measurable.

To see this suppose that ξ = [x_{1}] − [x_{2}] with x_{i} ∈ P (s) and ξ_{2} = [y_{1}] − [y_{2}] with y_{i} ∈ P (t). By replacing y_{1}, y_{2} with y_{1}v, y_{2}v for an appropriate v ∈ P we may assume that t is as large as we please, and in particular we may assume that t > s + 1. Choosing elements f ∈ P (t − s − λ) and e ∈ P (λ), we have

U_{λ}([x_{1}] − [x_{2}]) = U_{λ}([x_{1}f] − [x_{2}f]) = [ex_{1}f] − [ex_{2}f],

and thus

(4.11)  ⟨U_{λ}ξ_{1}, ξ_{2}⟩ = ⟨[ex_{1}f] − [ex_{2}f], [y_{1}] − [y_{2}]⟩ = α_{11} − α_{12} − α_{21} + α_{22},

where

α_{ij} = g(ex_{i}f, y_{j}).

In order to calculate the terms α_{ij} we make use of the propagators of y_{2} and y_{2} to obtain the factorizations

y_{j} = y_{j}(0, λ)y_{j}(λ, λ + t) = y_{j}(0, λ)y_{j}(λ, λ + s)y_{j}(λ + s, t).

Thus

α_{ij} =g(ex_{i}f, y_{j}(0, λ)y_{j}(λ, λ + t))
=g(e, y_{j}(0, λ)) + g(x_{i}f, y_{j}(λ, λ + t)) + ψ(e, x_{i}f) + \overline{ψ(y_{j}(0, λ), y_{j}(λ, λ + t))}
=g(e, y_{j}(0, λ)) + g(x_{i}, y_{j}(λ, λ + s)) + g(f, y_{j}(λ + s, t)) +

ψ(x_{i}, f) + \overline{ψ(y_{j}(λ, λ + s), y_{j}(λ + s, t))} + ψ(e, x_{i}f) + \overline{ψ(y_{j}(0, λ), y_{j}(λ, λ + t))}.

Noting that α_{ij} has the form

α_{ij} = g(x_{i}, y_{j}(λ, λ + s)) + u_{i} + v_{j}

for appropriate complex numbers u_{1}, u_{2}, v_{1}, v_{2} (which depend on λ), it follows that the u’s and v’s cancel out of the right side of (4.11) and we are left with

α_{11} − α_{12} − α_{21} + α_{22} =g(x_{1}, y_{1}(λ, λ + s)) − g(x_{2}, y_{1}(λ, λ + s))+
g(x_{2}, y_{2}(λ, λ + s)) − g(x_{1}, y_{2}(λ, λ + s)).

Since g is a measurable form, each of the two functions

λ ∈ (0, 1] ↦ g(x_{1}, y_{j}(λ, λ + s)) − g(x_{2}, y_{j}(λ, λ + s))

j = 1, 2 is a Borel function, and thus the right side of the previous formula is a difference of Borel functions.

Purity of {U_{t} : t ≥ 0}. We claim next that the semigroup {U_{t} : t ≥ 0} is pure in the sense that

(4.12)  ⋂_{t>0} U_{t}H_{∞} = {0}.

This is a consequence of (4.7.1), (4.7.2) and the following.

Lemma 4.13. For every t > 0, H_{∞} decomposes into a direct sum

H_{∞} = N_{t} ⊕ U_{t}H_{∞}.

proof. Fix t > 0. We show first that N_{t} is orthogonal to U_{t}H_{∞}. For that, it suffices to show that for every r > 0 and for x_{1}, x_{2} ∈ P (t), y_{1}, y_{2} ∈ P (r) we have

(4.14)  ⟨[x_{1}] − [x_{2}], U_{t}([y_{1}] − [y_{2}])⟩ = 0.

Choose elements e ∈ P (t), f ∈ P (r). Then we have

[x_{1}] − [x_{2}] = [x_{1}f] − [x_{2}f],  and
U_{t}([y_{1}] − [y_{2}]) = [ey_{1}] − [ey_{2}].

Thus the left side of (4.14) has the form

⟨[x_{1}f] − [x_{2}f], [ey_{1}] − [ey_{2}]⟩ = α_{11} − α_{12} − α_{21} + α_{22},

where

α_{ij} = g(x_{i}f, ey_{j}).

Using the definition of additive forms (2.3) we have

α_{ij} = g(x_{i}, e) + g(f, y_{j}) + ψ(x_{i}, f) + \overline{ψ(e, y_{j})} = u_{i} + v_{j}

where

u_{i} =g(x_{i}, e) + ψ(x_{i}, f)
v_{j} =g(f, y_{j}) + \overline{ψ(e, y_{j})}.

It follows that all of the u’s and v’s cancel and we are left with the required formula

α_{11} − α_{12} − α_{21} + α_{22} = 0.

To show that N_{t} ∪ U_{t}H_{∞} spans H_{∞} it is enough to show that for every r > t and every pair x_{1}, x_{2} ∈ P (r), we have

[x_{1}] − [x_{2}] ∈ N_{t} + U_{t}H_{∞}.

If we factor x_{i} = a_{i}b_{i} where a_{i} = x_{i}(0, t) ∈ P (t) and b_{i} = x_{i}(t, r) ∈ P (r − t), then we have

[x_{1}] − [x_{2}] = [a_{1}b_{1}] − [a_{2}b_{2}] =([a_{1}b_{1}] − [a_{2}b_{1}]) + ([a_{2}b_{1}] − [a_{2}b_{2}])
=([a_{1}] − [a_{2}]) + U_{t}([b_{1}] − [b_{2}])

and the right side clearly belongs to N_{t} + U_{t}H_{∞} □ {U_{t} : t ≥ 0} as a shift. A familiar theorem asserts that every strongly continuous pure semigroup of isometries is unitarily equivalent to a direct sum of copies of the semigroup of simple unilateral shifts acting on L^{2}(0, ∞). From this it follows that we can replace H_{∞} with the Hilbert space L^{2}((0, ∞); C) of all square integrable vector valued measurable functions ξ : (0, ∞) → C with inner product

⟨ξ, η⟩ = ∫_{0}^{∞} ⟨ξ(x), η(x)⟩ dx

in such a way that {U_{t} : t ≥ 0} becomes the semigroup

U_{t}ξ(x) = \{ \begin{matrix}ξ(x − t), & x > t \\ 0, & 0 < x ≤ t. \end{matrix}

After making this identification, we find that the range of U_{t} consists of all functions ξ ∈ L^{2}((0, ∞); C) which vanish almost everywhere in the interval 0 < x ≤ t. From Lemma 4.13 we conclude that for every t > 0,

(4.14)  \overline{span}\{[x_{1}] − [x_{2}] : x ∈ P (t)\} = L^{2}((0, t); C),

the right side denoting the subspace of all functions ξ ∈ L^{2}((0, ∞); C) which vanish almost everywhere outside the interval 0 < x ≤ t. Since H_{∞} is a separable Hilbert space, it follows that C must be separable as well.

Finally, we remind the reader that the rules for left and right multiplication in these new “coordinates” are the same as they were in H_{∞}: if x_{1}, x_{2} ∈ P (s) and u is any element of P (t), then

[x_{1}u] − [x_{2}u] = [x_{1}] − [x_{2}],  and
[ux_{1}] − [ux_{2}] = U_{t}([x_{1}] − [x_{2}]).

Equation (4.14) identifies the space N_{t} = \overline{span}\{[x_{1}] − [x_{2}] : x_{i} ∈ P (t)\} with the space P_{C}(t) of example 2.9 for every t > 0. It remains to define the “logarithm” mapping log : P → P_{C} with the properties asserted in Theorem 4.3.

Definition of the logarithm. In order to define the logarithm we must first show that certain 2-cocycles are trivial. These cocycles are associated with globally defined left-coherent sections

t ∈ (0, ∞) ↦ e_{t} ∈ P (t)

whose existence is established in the following.

Lemma 4.15. Choose any element e ∈ P (1). Then there is a left-coherent family of elements {e_{t} : t > 0} with the property that e_{1} = e.

proof. For 0 < t ≤ 1 we set e_{t} = e(0, t) where {e(s, t) : 0 ≤ s < t ≤ 1} is the propagator associated with e as in section 1. For n < t ≤ n + 1 we set

e_{t} = e_{1}^{n}e(0, t − n).

It is clear that {e_{t} : t > 0} has the required properties □

Choose such a section {e_{t} : t > 0}, which will be fixed throughout the remainder of this section. Define a function Γ : (0, ∞) × (0, ∞) → L^{2}((0, ∞); C) by

Γ(s, t) = [e_{s}e_{t}] − [e_{s+t}].

We will see presently that Γ is an additive 2-cocycle in the sense that for all r, s, t > 0 we have

Γ(r + s, t) − Γ(r, s + t) − U_{r}Γ(s, t) + Γ(r, s) = 0,

see (4.20). The following asserts that Γ is exact in the sense that we require.

Theorem 4.16. There is a measurable function t ∈ (0, ∞) ↦ ϕ_{t} ∈ L^{2}((0, ∞); C) such that

(4.16.1)  ϕ_{t}(x) =0  a.e. outside 0 < x ≤ t,
(4.16.2)  Γ(s, t) =ϕ_{s+t} − ϕ_{s} − U_{s}ϕ_{t},  s, t > 0.

Remark 4.17. The measurability assertion of Theorem 4.16 simply means that for every ξ ∈ L^{2}((0, ∞); C),

t ∈ (0, ∞) ↦ ⟨ϕ_{t}, ξ⟩

is a complex-valued Borel function. Because L^{2}((0, ∞); C) is separable, this is equivalent to measurability of t ↦ ϕ_{t} relative to the metric topology of L^{2}((0, ∞); C). Notice too that (4.16.1) asserts that ϕ_{t} ∈ P_{C}(t) for every t > 0.

Assume, for the moment, that Theorem 4.16 has been proved. We can then define a fiber map

log : P → P_{C}

as follows. For z ∈ P (t), t > 0 we put

log(z) = [z] − [e_{t}] − ϕ_{t}.

Notice that log(z) ∈ P_{C}(t) because both [z] − [e_{t}] and ϕ_{t} belong to P_{C}(t). For x, y ∈ P we claim:

(4.18)  log(xy) = log(x) ⊞ log(y).

This is to say that, if x ∈ P (s) and y ∈ P (t) then

log(xy) = log(x) + U_{s}(log(y));

equivalently,

[xy] − [e_{s+t}] − ϕ_{s+t} = [x] − [e_{s}] − ϕ_{s} + U_{s}([y] − [e_{t}] − ϕ_{t}).

To see that this is the case, note that

[xy] − [e_{s+t}] =([xy] − [e_{s}y]) + ([e_{s}y] − [e_{s}e_{t}]) + ([e_{s}e_{t}] − [e_{s+t}])
=[x] − [e_{s}] + U_{s}([y] − [e_{t}]) + Γ(s, t).

Using (4.16.2) to substitute for Γ(s, t) and subtracting ϕ_{s+t} from both sides, we obtain (4.18).

proof of Theorem 4.16. The argument will proceed as follows. We first find a family {u_{t} : t > 0} of Borel functions

u_{t} : (0, ∞) → C

which satisfy

Γ(s, t) = u_{s+t} − u_{s} − U_{s}(u_{t})

almost everywhere, for every s, t > 0. The family {u_{t} : t > 0} will not satisfy (4.16.1), but these functions will be locally in L^{2} in the sense that

∫_{0}^{T} ∥u_{t}(x)∥^{2} dx < ∞

for every T > 0. We will then use the results of section 3 to find a locally L^{2} function w : (0, ∞) → C with the property that for every t > 0

u_{t}(x + t) = w(x + t) − w(x)

almost everywhere (dx) on the interval (0, ∞). Such a function w can be subtracted from u_{t}

ϕ_{t}(x) = \{ \begin{matrix}u_{t}(x) − w(x), & 0 < x ≤ t \\ 0, & x > t \end{matrix}

so that the modification {ϕ_{t} : t > 0} satisfies both (4.16.1) and (4.16.2).

Lemma 4.19. For each s, t > 0, Γ(s, t)(x) vanishes almost everywhere (dx) outside the interval 0 < x < s + t. Γ is a 2-cocycle in the sense that for every r, s, t > 0 we have

(4.20)  Γ(r + s, t) − Γ(r, s + t) − U_{r}Γ(s, t) + Γ(r, s) = 0.

proof. Since both e_{s}e_{t} and e_{s+t} belong to P (s + t), Γ(s, t) = [e_{s}e_{t}] − [e_{s+t}] belongs to

\overline{span}\{[x_{1}] − [x_{2}] : x_{i} ∈ P (s + t)\} = L^{2}((0, s + t); C)\}.

Moreover, since e_{s+t} = e_{s}e(s, s + t), we see that

Γ(s, t) = [e_{s}e_{t}] − [e_{s}e(s, s + t)] = U_{s}([e_{t}] − [e(s, s + t)])

belongs to the range of U_{s} and hence Γ(s, t) must vanish almost everywhere on the interval 0 < x ≤ s.

To prove (4.20), notice that

U_{r}Γ(s, t) = U_{r}([e_{s}e_{t}] − [e_{s+t}]) = [e_{r}e_{s}e_{t}] − [e_{r}e_{s+t}],

and hence by the definition of Γ we have

Γ(r + s, t) − Γ(r, s + t) − U_{r}Γ(s, t)
= [e_{r+s}e_{t}] − [e_{r+s+t}] − [e_{r}e_{s+t}] + [e_{r+s+t}] − [e_{r}e_{s}e_{t}] + [e_{r}e_{s+t}]
= [e_{r+s}e_{t}] − [e_{r}e_{s}e_{t}] = [e_{r+s}] − [e_{r}e_{s}] = −Γ(r, s),

as required □ Lemma 4.21. For fixed t > 0 and ξ ∈ L^{2}((0, ∞); C), the function

s ↦ ⟨Γ(s, t), ξ⟩

is Borel-measurable.

proof. Fix t. Noting that

Γ(s, t) = [e_{s}e_{t}] − [e_{s+t}] = [e_{s}e_{t}] − [e_{s}e(s, s + t)]
= U_{s}([e_{t}] − [e(s, s + t)]),

we have

⟨Γ(s, t), ξ⟩ = ⟨[e_{t}] − [e(s, s + t)], U_{s}^{∗}ξ⟩ ,

for every s, t > 0. Since s ↦ U_{s}ξ is (metrically) continuous, it suffices to show that the function s ∈ (0, ∞) ↦ [e_{t}] − [e(s, s + t)] is weakly measurable; i.e., that

s ∈ (0, ∞) ↦ ⟨[e_{t}] − [e(s, s + t)], η⟩

is a Borel function for every η ∈ L^{2}((0, ∞); C). Since L^{2}((0, T); C) is spanned by {[y_{1}] − [y_{2}] : y_{i} ∈ P (T)} for every T > 0, this reduces to showing that for fixed T and y_{1}, y_{2} ∈ P (T),

s ∈ (0, ∞) ↦ ⟨[e_{t}] − [e(s, s + t)], [y_{1}] − [y_{2}]⟩

is a Borel function. To see that, pick u ∈ P (T −t). Then we have [e_{t}]−[e(s, s+t)] = [e_{t}u] − [e(s, s + t)u], and if we set x_{1} = e_{t}u and x_{2} = e(s, s + t)u then x_{i} ∈ P (T) and

⟨[e_{t}] − [e(s, s + t)], [y_{1}] − [y_{2}]⟩ = ⟨[x_{1}] − [x_{2}], [y_{1}] − [y_{2}]⟩
=α_{11} − α_{12} − α_{21} + α_{22},

where α_{ij} = g(x_{i}, y_{j}). We have

α_{1k} =g(e_{t}u, y_{k}),
α_{2k} =g(e(s, s + t)u, y_{k})

for k = 1, 2. By the additivity property 2.3 we have

α_{2k} =g(e(s, s + t)u, y_{k}) = g(e(s, s + t)u, y(0, t)y_{k}(t, T))
=g(e(s, s + t), y_{k}(0, t)) + g(u, y_{k}(t, T)) + ψ(e(s, s + t), u)
+ \overline{ψ(y_{k}(0, t), y_{k}(t, T))}.

Noting that neither α_{11} nor α_{12} involves s and that the terms ψ(e(s, s+t), u) cancel out of the difference α_{22} − α_{21}, we find that

α_{11} − α_{12} − α_{21} + α_{22} = −g(e(s, s + t), y_{1}(0, t)) + g(e(s, s + t), y_{2}(0, t)) + K

where K does not depend on s. The right side is a Borel function of s because of the measurability hypothesis on g □

We define a family of functions u_{s} : (0, ∞) → C as follows.

Lemma 4.22. For every s > 0, the limit

(4.22.1)  u_{s}(λ) = − lim_{n→∞} Γ(s, n)(λ)

exists almost everywhere on 0 < λ < ∞ and satisfies

(4.22.2)  ∫_{0}^{T} ∥u_{s}(λ)∥^{2} dλ < ∞

for every T > 0. {u_{s} : s > 0} is measurable in the sense that for every compactly supported ξ ∈ L^{2}((0, ∞); C), the function

s ∈ (0, ∞) ↦ ⟨u_{s}, ξ⟩

is Borel measurable. Putting u_{s}(λ) = 0 for λ ≤ 0 we have

(4.22.3)  Γ(s, t)(λ) = u_{s+t}(λ) − u_{s}(λ) − u_{t}(λ − s)

almost everywhere on 0 < λ < ∞, for every s, t > 0.

Remark. Actually, the limit in (4.22.1) exists in a very strong sense. We will show that as t increases with s fixed, the restrictions

Γ(s, t) ↾_{(0,T]}

stabilize as soon as t is larger than T. Once one knows this, the assertion (4.22.2) is an obvious consequence of the fact that each function Γ(s, t) belongs to L^{2}(0, ∞); C).

proof of Lemma 4.22. We first establish the coherence property described in the preceding remark. More precisely, we claim that for fixed 0 < s < T < t_{1} < t_{2} one has

(4.23)  Γ(s, t_{2}) ↾_{(0,T]}= Γ(s, t_{1}) ↾_{(0,T]} .

To see that, consider the difference

Γ(s, t_{2}) − Γ(s, t_{1}) = [e_{s}e_{t_{2}}] − [e_{s+t_{2}}] − [e_{s}e_{t_{1}}] + [e_{s+t_{1}}].

Writing

[e_{s}e_{t_{2}}] − [e_{s}e_{t_{1}}] = [e_{s}e_{T −s}e(T − s, t_{2})] − [e_{s}e_{T −s}e(T − s, t_{1})]
= U_{T} ([e(T − s, t_{2})] − [e(T − s, t_{1})])

and noting that

−[e_{s+t_{2}}] + [e_{s+t_{1}}] = −[e_{T} e(T, t_{2})] + [e_{T} e(T, t_{1})]
= −U_{T} ([e(T, t_{2})] − [e(T, t_{1})]

we find that Γ(s, t_{2}) − Γ(s, t_{1}) has the form U_{T} ζ for ζ ∈ L^{2} given by

ζ = [e(T − s, t_{2})] − [e(T − s, t_{1})] − [e(T, t_{2})] + [e(T, t_{1})].

(4.23) follows because every function in the range of U_{T} vanishes a.e. on the interval (0, T].

Thus (4.22.1) follows and by the preceding remark we also have (4.22.2). It is also clear that for every compactly supported function ξ ∈ L^{2},

⟨u_{s}, ξ⟩ = ⟨Γ(s, n), ξ⟩

for sufficiently large n = 1, 2, . . .. Thus the measurability of {u_{s}} follows as well.

Finally, the formula (4.22.3) follows after restricting all terms in the cocycle equation (4.16.2) to a finite unterval 0 < λ ≤ T and taking the formal lim_{t→∞} to obtain

−u_{r+s} + u_{r} + U_{r}u_{s} + Γ(r, s+ = 0,

for every r, s > 0 □

We must now modify the family {u_{s} : s > 0} in order to obtain a new family ϕ_{s} = u_{s} − w which has the additional property that ϕ_{s}(s) vanishes a.e. outside the interval 0 < λ ≤ s. This is accomplished as follows.

Notice that for s, t > 0,

u_{s+t}(λ) = u_{s}(λ) + u_{t}(λ − s)

almost everywhere on the interval λ ≥ s + t. Indeed, this is immediate from the fact that

Γ(s, t)(λ) = u_{s+t}(λ) − u_{s}(λ) − u_{t}(λ − s)

and the fact that Γ(s, t) vanishes outside the interval 0 < λ ≤ s + t. Thus if we define v_{t} : (0, ∞) → C by

v_{t}(λ) = u_{t}(λ + t),

then {v_{t} : t > 0} is a measurable family of C-valued functions satisfying

∫_{0}^{T} ∥v_{t}(λ)∥^{2} dλ < ∞

for every T, t > 0, for which

v_{s+t}(λ) = v_{s}(λ) + v_{t}(λ + s)

almost everywhere (dλ), for every s, t > 0. By Theorem 3.2, there is a Borel function

w : (0, ∞) → C

which is locally in L^{2}, such that for every t > 0 we have

v_{t}(λ) = w(λ + t) − w(λ)

almost everywhere on the interval 0 < λ < ∞. Set w(λ) = 0 for λ ≤ 0. It follows that

u_{t}(λ) − w(λ) + w(λ − t)

vanishes almost everywhere on the interval t < λ < ∞. Hence

ϕ_{t}(λ) = u_{t}(λ) − w(λ) + w(λ − t)

satisfies both conditions

ϕ_{s+t}(λ) − ϕ_{s}(λ) − ϕ_{t}(λ − s) = Γ(s, t)(λ)

a.e. on 0 < λ < ∞ and

ϕ_{t}(λ) = 0

almost everywhere on the interval λ > t. Notice that we can also define ϕ_{t} as follows

(4.24)  ϕ_{t}(λ) = \{ \begin{matrix}u_{t}(λ) − w(λ), & 0 < λ ≤ t \\ 0, & λ > t. \end{matrix}

As in the remarks following the statement of Theorem 4.16, we can now define a fiber map log : P → P_{C} by

log(x) = [x] − [e_{t}] − ϕ_{t}

for every x ∈ P (t), and every t > 0, and this function satisfies log(P (t)) ⊆ P_{C}(t) and log(xy) = log(x) ⊞ log(y), for x, y ∈ P.

It remains to establish (4.3.1), and to exhibit a function ρ : P → ℂ which satisfies (4.3.2). ρ is defined as follows. If x ∈ P (t) we put

ρ(x) = ⟨[x] − [e_{t}], ϕ_{t}⟩ + g(x, e_{t}) − \frac{1}{2}(g(e_{t}, e_{t}) + ∥ϕ_{t}∥^{2}).

To see that (4.3.2) is satisfied we choose x_{1}, x_{2} ∈ P (t) and use the definition of log to write

⟨log(x_{1}), log(x_{2})⟩ = ⟨[x_{1}] − [e_{t}] − ϕ_{t}, [x_{2}] − [e_{t}] − ϕ_{t}⟩
= ⟨[x_{1}] − [e_{t}], [x_{2}] − [e_{t}]⟩ − ⟨[x_{1}] − [e_{t}], ϕ_{t}⟩
− ⟨ϕ_{t}, [x_{2}] − [e_{t}]⟩ − ∥ϕ_{t}∥^{2}.

Noting that ⟨[x_{1}] − [e_{t}], [x_{2}] − [e_{t}]⟩ expands to

g(x_{1}, x_{2}) − g(x_{1}, e_{t}) − g(e_{t}, x_{2}) + g(e_{t}, e_{t}),

we obtain the required formula

⟨log(x_{1}), log(x_{2})⟩ = g(x_{1}, x_{2}) + ρ(x_{1}) + \overline{ρ(x_{2})}.

It remains to show that log(P (t)) is a strongly spanning set in P_{C}(t). To see that, fix t > 0, and let us write L = log(P (t)). Notice that L carries no linear structure a priori, since the only algebraic property of the log function is its additivity

log(xy) = log(x) ⊞ log(y).

Nevertheless, we will show that L is “almost convex”.

For every r > 0, let

B_{r} = {ξ ∈ P_{C}(t) : ∥ξ∥ ≤ r}

be the ball of radius r and let

K = ⋃_{r>0} \overline{L ∩ B_{r}}^{w}

where \overline{L ∩ B_{r}}^{w} denotes the closure of L ∩ B_{r} in the weak topology of the Hilbert space P_{C}(t).

Lemma 4.26. If ξ, η ∈ K and θ is a dyadic rational in the unit interval, then θξ + (1 − θ)η ∈ K.

proof. It clearly suffices to prove that

ξ, η ∈ K ⟹ \frac{1}{2}(ξ + η) ∈ K.

We will show first that this is true in the special case where ξ, η have the form

ξ = log(x), η = log(y)

with x, y ∈ P (t). To this end we claim that there is a sequence z_{n} ∈ P (t) with the properties

(4.27)  ∥ log(z_{n})∥ ≤ ∥ log(x)∥ + ∥ log(y)∥

and which, in addition, satisfies

lim_{n→∞} ⟨log(z_{n}), ζ⟩ = \frac{1}{2} ⟨log(x), ζ⟩ + \frac{1}{2} ⟨log(y), ζ⟩ ,

for all ζ ∈ P_{C}(t). Indeed, for every n = 1, 2, …, consider a dyadic partition of the interval [0, t] as follows,

{0 = t_{0} < t_{1} < ⋯ < t_{2^{n}} = t}

where t_{k} = kt/2^{n}, 0 ≤ k ≤ 2^{n}. Using the propagators {x(r, s) : 0 ≤ r < s ≤ t} and {y(r, s) : 0 ≤ r < s ≤ t} for x and y we can define z_{n} as a product

z_{n} = x_{1}y_{2}x_{3}y_{4} … x_{2^{n}−1}y_{2^{n}}

where x_{k} = x(t_{k−1}, t_{k}) and y_{k} = y(t_{k−1}, t_{k}). Then because of the additivity property of log we have

log(z_{n}) = log(x_{1}) ⊞ log(y_{2}) ⊞ log(x_{3}) ⊞ log(y_{4}) ⊞ ⋯ ⊞ log(x_{2^{n}−1}) ⊞ log(y_{2^{n}}).

Letting O_{n} and E_{n} be the respective unions of the odd and even intervals,

O_{n} = ⋃_{k odd}(t_{k−1}, t_{k}]
E_{n} = ⋃_{k even}(t_{k−1}, t_{k}]

we can rewrite the previous formula for log(z_{n}) as follows

log(z_{n}) = log(x)χ_{O_{n}} + log(y)χ_{E_{n}},

χ_{S} denoting the characteristic function of the set S ⊆ [0, t]. It follows that

∥ log(z_{n})∥ ≤ ∥ log(x)∥ + ∥ log(y)∥.

Moreover, the arument of [2, p. 47] implies that for any function w in L^{1}[0, t] we have

lim_{n→∞} ∫_{O_{n}} w(x) dx = \frac{1}{2} ∫_{0}^{t} w(x) dx  and
lim_{n→∞} ∫_{E_{n}} w(x) dx = \frac{1}{2} ∫_{0}^{t} w(x) dx

(An equivalent assertion is that the sequence χ_{E_{n}} ∈ L^{∞}[0, t] converges to the constant function 1/2 in the weak^{∗}-topology of L^{∞}[0, t]). So if we fix ζ ∈ P_{C}(t) = L^{2}((0, t); C) then we have

⟨log(z_{n}), ζ⟩ = ⟨log(x) · χ_{O_{n}} + log(y)χ_{E_{n}}, ζ⟩ = ∫_{O_{n}} w_{1}(λ) dλ + ∫_{E_{n}} w_{2}(x) dx,

where w_{1}(λ) = ⟨log(x)(λ), ζ(λ)⟩, w_{2}(λ) = ⟨log(y)(λ), ζ(λ)⟩. By the preceding remarks the right side tends in the limit to

\frac{1}{2} ∫_{0}^{t} ⟨log(x)(λ), ζ(λ)⟩ dλ + \frac{1}{2} ∫_{0}^{t} ⟨log(y)(λ), ζ(λ)⟩ dλ

as asserted.

Now let ξ, η be arbitrary elements of K. Since the sets \overline{L ∩ B_{r}}^{w} increase with r, we can assume that

ξ, η ∈ \overline{L ∩ B_{r}}^{w}

for some r > 0. It follows that there are sequences x_{n}, y_{n} ∈ P (t) satisfying ∥ log(x_{n})∥ ≤ r, ∥ log(y_{n})∥ ≤ r and

ξ = lim_{n→∞} log(x_{n}),
η = lim_{n→∞} log(y_{n})

weakly. Now for each fixed n = 1, 2, … the preceding argument implies that

\frac{1}{2} log(x_{n}) + \frac{1}{2} log(y_{n}) ∈ \overline{L ∩ B_{2r}}^{w}.

Since the set on the right is weakly closed, we may take the limit on n to obtain

\frac{1}{2}ξ + \frac{1}{2}η ∈ \overline{L ∩ B_{2r}}^{w},

as required □

Remark 4.29. From Lemma 4.26 we immediately deduce: the norm closure of

K = ⋃_{r>0} \overline{log(P (t)) ∩ B_{r}}^{w}

is a convex subset of P_{C}(t).

Lemma 4.30. Set S be a convex subset of a Hilbert space H which spans H. Then S is a strongly spanning set.

proof. If S contains 0 then the conclusion follows from [2 Proposition 6.12]. Thus we will obtain the more general result if we show that for every strongly spanning set S_{0} ⊆ H and every ξ ∈ H, S_{0} + ξ is also a strongly spanning set.

To see that, let ζ ∈ H ↦ W_{ζ} ∈ U(B(e^{H})) be the standard representation of the canonical commutation relations on the symmetric Fock space e^{H}; W_{ζ} is defined by requiring

W_{ζ} : exp(η) ↦ e^{− \frac{1}{2} ∥η∥^{2}−⟨η,ζ⟩} exp(ζ + η),

for every η ∈ H. Each W_{ζ} is a unitary operator on e^{H}. Now let v be a vector in e^{H} such that ⟨v, exp(η)⟩ = 0 for every η ∈ S_{0} + ξ. We have to show that v = 0. But for every η_{0} ∈ S_{0} we have

⟨v, W_{ξ}(exp(η_{0}))⟩ = e^{− \frac{1}{2} ∥η∥^{2}−⟨ζ,η⟩} ⟨v, exp(η_{0} + ξ)⟩ = 0

for every η_{0} ∈ S_{0}, hence W_{ξ}^{∗}v is orthogonal to exp(S_{0}). Since exp(S_{0}) spans e^{H} we conclude that W_{ξ}^{∗}v = 0, hence v = 0 □

We can now show that L = log(P (t)) is a strongly spanning subset of P_{C}(t). To see that, note first that the exponential map exp : ξ ↦ exp(ξ) is weakly continuous on bounded subsets of P_{C}(t). Indeed, if {ξ_{α}} is a bounded net in P_{C}(t) which converges weakly to ξ_{∞}, then for every η ∈ P_{C}(t) we have

lim_{α} ⟨exp(ξ_{α}), exp(η)⟩ = lim_{α} e^{⟨ξ_{α},η⟩} = e^{⟨ξ_{∞},η⟩}
= ⟨exp(ξ_{∞}), exp(η)⟩ .

Since exp(ξ_{α}) is a bounded net in the symmetric Fock space over P_{C}(t) and since the set of vectors {exp(η) : η ∈ P_{C}(t)} span this space, it follows that exp(ξ_{α}) converges weakly to exp(ξ_{∞}).

Now choose a vector v in the symmetric Fock space over P_{C}(t) with the property that ⟨v, exp(L)⟩ = {0}. The preceding paragraph implies that v is orthogonal to the set of vectors exp(\overline{L ∩ B_{r}}^{w}) for every r > 0, and taking the union over r > 0 we obtain ⟨v, exp(K)⟩ = {0}. Since the exponential map exp is metrically continuous on its entire domain P_{C}(t), it follows that v is orthogonal to the set of vectors exp(\overline{K}), \overline{K} denoting the closure of K in the norm topology of P_{C}(t). By remark 4.29 and Lemma 4.30, we conclude that v = 0.

That completes the proof of Theorem 4.3 □

Classification of multiplicative forms. We conclude this section with a discussion of how Theorem 4.3 gives a classification of multiplicative structures on path spaces. Let (P, g) be a metric path space and let e^{g} : P^{2} → ℂ be its associated multiplicative form, defined on P (t) × P (t) for t > 0 by

x, y ∈ P (t) ↦ e^{g(x,y)}.

e^{g} is a positive definite function on P (t) × P (t) and hence there is a Hilbert space E(t) and a function F_{t} : P (t) → E(t) satisfying

E(t) =\overline{span}\{F_{t}(x) : x ∈ P (t)\}
⟨F_{t}(x), F_{t}(y)⟩ =e^{g(x,y)},  x, y ∈ P (t).

Thus we have a family of Hilbert spaces p : E → (0, ∞)

E = {(t, ξ) : t > 0, ξ ∈ E(t)}

with projection p(t, ξ) = t.

We define a binary operation ξ, η ∈ E ↦ ξ · η ∈ E as follows. Since g is additive there is a function ψ : P × P → ℂ such that for all x_{1}, x_{2} ∈ P (s), y_{1}, y_{2} ∈ P (t)

g(x_{1}y_{1}, x_{2}y_{2}) = g(x_{1}, x_{2}) + g(y_{1}, y_{2}) + ψ(x_{1}, y_{1}) + \overline{ψ(x_{2}, y_{2})}.

For x ∈ P (s), y ∈ P (t) we try to define the product F_{s}(x) · F_{t}(y) by

(4.31)  F_{s}(x) · F_{t}(y) = e^{−ψ(x,y)}F_{s+t}(xy).

It follows that for x_{1}, x_{2} ∈ P (s), y_{1}, y_{2} ∈ P (t) we have

⟨F_{s}(x_{1}) · F_{t}(y_{1}), F_{s}(x_{2}) · F_{t}(y_{2})⟩ = e^{g(x_{1}y_{1},x_{2}y_{2})−ψ(x_{1},y_{1})−\overline{ψ(x_{2},y_{2})}} = e^{g(x_{1},x_{2})+g(y_{1},y_{2})}
= ⟨F_{s}(x_{1}), F_{s}(x_{2})⟩ ⟨F_{t}(y_{1}), F_{t}(y_{2})⟩ .

The latter formula implies that there is a unique unitary operator

W_{s,t} : E(s) ⊗ E(t) → E(s + t)

satisfying

W_{s,t}(F_{s}(x) ⊗ F_{t}(y)) = F_{s}(x) · F_{t}(y).

Thus we can define a bounded bilinear map (ξ, η) ∈ E(s) × E(t) ↦ ξ · η ∈ E(s + t) by way of

ξ · η = W_{s,t}(ξ ⊗ η)

and this mapping extends the operation (4.31).

To see that this operation on E is associative, it suffices to show that it is associative on generators, i.e.,

F_{r}(x) · (F_{s}(y) · F_{t}(z)) = (F_{r}(x) · F_{s}(y)) · F_{t}(z)

for all x ∈ P (r), y ∈ P (s), z ∈ P (t). Using the definition (4.31), one observes that this will follow provided that ψ satisfies

(4.32)  ψ(x, y) + ψ(xy, z) = ψ(x, yz) + ψ(y, z).

In fact, the equation (4.32) can be arranged a priori from the definition of ψ (2.4). But it is easier at this point to invoke Theorem 4.3. The latter asserts that there is a complex-valued function ρ defined on P such that

ψ(x, y) = ρ(xy) − ρ(x) − ρ(y)

for all x, y ∈ P. Substituting this into (4.32) one finds (using associativity of the multiplication in P) that both sides of (4.32) reduce to

ρ(xyz) − ρ(x) − ρ(y) − ρ(z).

This proves associativity of the multiplication in E.

The preceding discussion implies that this multiplication acts like tensoring. Thus we have a product structure satisfying all the axioms of a product system except measurability requirements. Using Theorem 4.3, we can describe this product structure as follows.

Corollary 4.33. Let (P, g) be a metric path space and let E be the product structure obtained from the positive definite functions

(x, y) ∈ P (t) × P (t) ↦ e^{g(x,y)},

t > 0. Assume that E(t) is not one-dimensional for every t. Then E is isomorphic to the product structure of one of the standard product systems E_{1}, E_{2}, …, E_{∞}.

proof. By Theorem 4.3, there is a separable Hilbert space C, a complex-valued function ρ : P → ℂ and a fiber map log : P → P_{C} satisfying the conditions (4.3.1) and (4.3.2). In view of the remarks following Theorem 4.3, we may take the defect to be of the form

ψ(x, y) = ρ(xy) − ρ(x) − ρ(y).

Using the formula (4.31) we find that if we rescale F_{t} : P (t) → E(t) according to

g_{t}(x) = e^{−ρ(x)}F_{t}(x),

then the definition of multiplication in E simplifies to

G_{s}(x) · G_{t}(y) = G_{s+t}(xy),

for x ∈ P (s), y ∈ P (t), and all s, t > 0.

Moreover, formula (4.3.2) implies that for x_{1}, x_{2} ∈ P (t) we have

⟨G_{t}(x_{1}), G_{t}(x_{2})⟩ = e^{g(x_{1},x_{2})−ρ(x_{1})−\overline{ρ(x_{2})}} = e^{⟨log(x_{1}),log(x_{2})⟩}.

Now consider the exponential map

exp : L^{2}((0, ∞); C) → e^{L^{2}((0,∞);C)}.

The latter formula asserts that

⟨G_{t}(x_{1}), G_{t}(x_{2})⟩ = ⟨exp(log(x_{1})), exp(log(x_{2}))⟩

for all x_{1}, x_{2} ∈ P (t). This implies that we can define an isometry

W_{t} : E(t) → e^{P_{C}(t)}

by way of

W_{t}(G_{t}(x)) = exp(log(x)),

for x ∈ P (t). By (4.3.1), each W_{t} is a unitary operator. The total map

W : E → e^{P_{C}}

is an isomorphism of families of Hilbert spaces.

It remains to verify that W preserves multiplication, i.e., that

W_{s+t}(ξ · η) = W_{s}(ξ)W_{t}(η)

for every ξ ∈ E(s), η ∈ E(t). Recalling that the multiplication in e^{P_{C}} is defined by

exp(f) exp(g) = exp(f ⊞ g)

for f ∈ P_{C}(s), g ∈ P_{C}(t) we find that for all x ∈ P (s), y ∈ P (t),

W_{s+t}(G_{s}(x) · G_{t}(y)) = W_{s+t}(G_{s+t}(xy)) = exp(log(xy))

= exp(log(x) ⊞ log(y)) = exp(log(x)) exp(log(y)),

and hence

W_{s+t}(G_{s}(x) · G_{t}(y)) = W_{s}(G_{s}(x))W_{t}(G_{t}(y)).

The assertion follows from the bilinearity of multiplication and the fact that E(r) is spanned by G_{r}(P (r)) for every r > 0

Finally, note that if the space C of “coordinates” is the trivial Hilbert space {0} then P_{C}(t) = L^{2}((0, t); C) is trivial as well and hence

e^{P_{C}(t)}

is one-dimensional for every t > 0. By virtue of the isomorphism W : E → e^{P_{C}}, this has been ruled out in the hypothesis of Corollary 4.33. Thus n = dim(C) is a positive integer or ℵ_{0}. In this case, W implements an isomorphism of the product structure E onto the product structure of E_{n} □

Part II. Continuous Tensor Products

Introduction to Part II.

Let p : E → (0, ∞) be a product system. Thus each fiber E(t) = p^{−1}(t) is a separable Hilbert space and we are given an associative multiplication (x, y) ∈ E × E ↦ xy ∈ E which acts like tensoring in the sense that for fixed s, t > 0,

(x, y) ∈ E(s) × E(t) ↦ xy ∈ E(s + t)

is a bilinear mapping with the properties

(II.1)  E(s + t) = \overline{span}E(s)E(t)
(II.2)  ⟨x_{1}y_{1}, x_{2}y_{2}⟩ = ⟨x_{1}, x_{2}⟩ ⟨y_{1}, y_{2}⟩ ,

for x_{i} ∈ E(s), y_{i} ∈ E(t). In addition, there are natural measurability axioms which we will not repeat here [2]. We will write E = {E(t) : t > 0} instead of p : E → (0, ∞) when it is convenient.

A nonzero vector x ∈ E(t) is called decomposable if for every 0 < s < t there are vectors y ∈ E(s), z ∈ E(t − s) for which

(II.3)  x = yz.

The set of decomposable vectors in E(t) will be written D(t).

There are product systems which contain no decomposable vectors at all. But if there is a t_{0} > 0 for which D(t_{0}) ≠ ∅, then D(t) ≠ ∅ for every t > 0 and we clearly have

D(s + t) = D(s)D(t)

for every s, t > 0. This multiplicatively structured family of sets {D(t) : t > 0} comes close to defining a path space, except that the factorizations of (II.3) are not unique. However, if y_{1}, y_{2} ∈ E(s) and z_{1}, z_{2} ∈ E(t − s) satisfy y_{1}z_{1} = y_{2}z_{2}, then because of the identification of E(t) with the tensor product E(s) ⊗ E(t − s) described in (II.1) and (II.2), we see that there must be a nonzero complex number λ such that

y_{2} = λy_{1}, z_{2} = λ^{−1}z_{1}.

Thus we may obtain a path space structure by passing from each D(t) to its associated projective space Δ(t).

More explictly, Δ(t) is obtained by identifying two vectors x_{1}, x_{2} in D(t) which are nonzero scalar multiples of each other. We consider Δ(t) to be a set with no additional structure. There is a natural projection x ∈ D(t) ↦ ẋ ∈ Δ(t). Any complex function f : D(t) → ℂ which is homogeneous of degree zero in the sense that f(λx) = f(x) for all nonzero scalars λ can be promoted to a function ḟ : Δ(t) → ℂ by way of ḟ(ẋ) = f(x), x ∈ D(t). In fact, it will be convenient to abuse notation slightly and identify functions on Δ(t) with homogenous functions defined on D(t). The path space p : Δ → (0, ∞) is defined by

Δ = {(t, ẋ) : t > 0, x ∈ D(t)},

with projection p(t, ẋ) = t and multiplication (s, ẋ)(t, ẏ) = (s + t, ẋy). We remind the reader that Δ, like any path space, is to be considered a fibered set with no additional structure beyond the multiplication it carries. Δ^{2} will denote the fiber product

Δ^{2} = {(t, ẋ, ẏ) : t > 0, ẋ, ẏ ∈ Δ(t)}.

For example, if for each t > 0 we are given a function f_{t} : D(t) → ℂ which satisfies f_{t}(λx) = f_{t}(x) for x ∈ D(t) and λ ≠ 0, then according to the abuses that have been agreed to we can define a function ϕ : Δ → ℂ by way of

ϕ(t, x) = f_{t}(x), x ∈ D(t).

Of course, the inner product restricts to a positive definite function on every D(t)

(x, y) ∈ D(t) × D(t) ↦ ⟨x, y⟩ ∈ ℂ,

but this function cannot be promoted to one defined on Δ(t) × Δ(t). We will see in section 6 below that the inner product of any two vectors in D(t) must be nonzero. Thus if we choose a fixed element e ∈ D(t) then we may form the renormalized inner product

P_{t}(x, y) = \frac{⟨x, y⟩}{⟨x, e⟩ ⟨e, y⟩}

and the latter function can be promoted to a positive definite function on Δ(t) × Δ(t). If we choose a family {e_{t} ∈ D(t) : t > 0} of decomposable vectors then we obtain a function

P : Δ^{2} → ℂ

which restricts to a positive definite function on Δ(t) × Δ(t) for every t > 0. Of course, these renormalized versions of the inner product depend on the particular choice of {e_{t} : t > 0}.

The results of sections 5–9 below combine to show that it is possible to find additive forms g : Δ^{2} → ℂ for this path space which are logarithms of the inner product in the following sense.

Theorem A. Let {e_{t} ∈ D(t) : t > 0} be a family of decomposable vectors which is left-coherent in the sense that for every s, t satisfying 0 < s < t there is a vector e(s, t) ∈ D(t − s) such that

e_{t} = e_{s}e(s, t).

Then there is an additive form g : Δ^{2} → ℂ (which will depend on e) such that for every t > 0 and every x_{1}, x_{2} ∈ D(t) we have

⟨x_{1}, x_{2}⟩ = ⟨x_{1}, e_{t}⟩ \overline{⟨x_{2}, e_{t}⟩}e^{g(x_{1},x_{2})}.

In fact, we will show that g : Δ^{2} → ℂ is a “continuous” logarithm which “vanishes at zero”; moreover, it is uniquely determined by these requirements once {e_{t}} is fixed. We emphasize that for typical choices of e the form g will have nonzero defect.

In sections 5–7, we establish certain continuity and nonvanishing properties of inner products of decomposable vectors. In section 8 we construct g as a “continuous” logarithm, and in section 9 we show that it has the required positivity properties.

5. Continuity of the modulus.

Let E = {E(t) : t > 0} be a product system. Suppose that we are given vectors x ∈ E(s), y ∈ E(t) with 0 < s < t. x is called a left divisor (resp. right divisor) of y if there is a vector z ∈ E(t − s) such that y = xz (resp. y = zx). Notice that in either case we have ∥x∥ · ∥z∥ = ∥y∥ and hence both x and z are nonzero whenever y is nonzero. Notice too that, while factorizations of the form II.3 are not unique, we do have both left and right cancellation laws. That is, if y ∈ E(s) and z_{1}, z_{2} ∈ E(t), then yz_{1} = yz_{2} ⟹ z_{1} = z_{2}, and z_{1}y = z_{2}y ⟹ z_{1} = z_{2}.

Given 0 < T ≤ ∞, we say that a family of vectors {x_{t} ∈ E(t) : 0 < t < T } is left-coherent (resp. right-coherent)if for every 0 < s_{1} < s_{2} < T, x_{s_{1}} is a left (resp. right) divisor of x_{s_{2}}. Our analysis is based on the following continuity property of coherent families of vectors. The proof makes use of a central technical result from [3].

Theorem 5.1. Let {x_{t} : 0 < t < T } and {y_{t} : 0 < t < T } be two left-coherent (resp. right-coherent) families of vectors satisfying ∥x_{t}∥ = ∥y_{t}∥ = 1 for all t. Then

lim_{t→0+} | ⟨x_{t}, y_{t}⟩ | = 1.

Remarks. Notice that the quantity | ⟨x_{t}, y_{t}⟩ | does not exceed 1 and increases as t decreases to 0. Indeed, if 0 < s < t < T then in the left-coherent case we can write x_{t} = x_{s}u, y_{t} = y_{s}v where u, v are unit vectors in E(t − s), hence

| ⟨x_{t}, y_{t}⟩ | = | ⟨x_{s}u, y_{s}v⟩ | = | ⟨x_{s}, y_{s}⟩ | · | ⟨u, v⟩ | ≤ | ⟨x_{s}, y_{s}⟩ | ≤ 1.

It follows that the essential assertion of Theorem 5.1 is that

sup_{0<t<T} | ⟨x_{t}, y_{t}⟩ | = 1.

Notice too that we cannot draw any conclusion about continuity of the inner product itself. Indeed, if we start with {y_{t}} as above and define x_{t} by x_{t} = f(t)y_{t} where f is an arbitrary function from (0, ∞) to the unit circle in the complex plane, then < x_{t}, y_{t} >= f(t) can be pathological.

proof of Theorem 5.1. We will prove Theorem 5.1 for left-coherent families. With that in hand, the corresponding conclusion for right-coherent families follows from it by considering the product system E^{o} opposite to E (see [7,8]). Moreover, by passing to subfamilies if necessary, we may assume that T is finite and positive.

For every s ∈ (0, T) we consider a projection P_{s} ∈ B(E(T)) which is defined as follows. Since ∥x_{s}∥ = 1, the operator

L_{x_{s}} : z ∈ E(T − s) → x_{s}z ∈ E_{T}

is an isometry whose range projection is given by

P s = L_{x_{s}}L_{x_{s}}^{∗} ∈ B(E(T)).

The action of P_{s} on a product vector of the form

y = y_{1}y_{2}, y_{1} ∈ E(s), y_{2} ∈ E(T − s)

is given by

P_{s}y = ⟨y_{1}, x_{s}⟩ x_{s}y_{2}.

Note that if 0 < s_{1} < s_{2} < T then P_{s_{1}} ≥ P_{s_{2}}, or equivalently that

(5.2)  x_{s_{2}}E(T − s_{2}) ⊆ x_{s_{1}}E(T − s_{1}).

Indeed, because of left-coherence x_{s_{2}} can be factored into a product x_{s_{2}} = x_{s_{1}}z where z ∈ E(s_{2} − s_{1}). Hence zE(T − s_{2}) ⊆ E(T − s_{1}), from which (5.2) is evident.

Therefore, as s decreases to 0 the projections P_{s} increase strongly to a limit projection P_{0}:

P_{0} = strong limit_{s→0+} P_{s}.

We claim that P_{0} = 1. Since P_{0} is clearly a nonzero projection in B(E(T)), it suffices to exhibit an irreducible ∗-algebra A of operators on E(T) which commutes with P_{0}. We can exhibit such an algebra as follows.

For every 0 < s < T we can identify E(T) with the tensor product E(s)⊗E(T −s) by associating a product xy ∈ E(T), x ∈ E(s), y ∈ E(T − s) with x ⊗ y ∈ E(s) ⊗ E(T − s). In this identification, P_{s} is associated with the projection

[x_{s}] ⊗ 1_{E(T −s)},

[x_{s}] denoting the rank-one projection ξ ∈ E(s) ↦ ⟨ξ, x_{s}⟩ x_{s} ∈ E(s). Thus P_{s} commutes with the von Neumann algebra

A_{s} = 1_{s} ⊗ B(E(T − s)).

As s decreases to 0 the operator algebras A_{s} increase, and hence P_{0} commutes with the union

A = ⋃_{0<s<T} A_{s}.

By Proposition 5.5 of [3], the union A is irreducible. Hence P_{0} = 1.

It follows that

lim_{s→0+} ∥P_{s}ξ∥ = ∥ξ∥,

for every vector ξ ∈ E(T). Taking ξ = y_{T} and noting that for every 0 < s < T, y factors into a product y_{T} = y_{s}z_{T −s} where z_{r} is a unit vector in E(r) for every 0 < r < T, we have

P_{s}y_{T} = P_{s}(y_{s}z_{T −s}) = ⟨y_{s}, x_{s}⟩ x_{s}z_{T −s}

and hence

1 = ∥y_{T} ∥ = lim_{s→0+} ∥P_{s}y_{T} ∥ = lim_{s→0+} | ⟨y_{s}, x_{s}⟩ | · ∥x_{s}∥ · ∥z_{T −s}∥ = lim_{s→0+} | ⟨y_{s}, x_{s}⟩ |,

as required □

With 5.1 in hand, we can now establish the following basic result on continuity of inner products.

Theorem 5.3. Let {x_{t} : 0 < t < T } and {y_{t} : 0 < t < T } be two families of vectors which satisfy the hypotheses of Theorem 5.1. Then the function

ϕ(t) = | ⟨x_{t}, y_{t}⟩ |

is continuous on the interval 0 < t < T and tends to 1 as t → 0+.

proof. As in the proof of 5.1, it suffices to consider the case where both families are left-coherent, and where ∥x_{t}∥ = ∥y_{t}∥ = 1 for all t. Choose t_{0} ∈ (0, ∞). We will show that ϕ is both left continuous and right continuous at t_{0}.

right continuity. For every λ ∈ (0, T − t_{0}) we may find u_{λ}, v_{λ} ∈ E(λ) such that

x_{t_{0}+λ} = x_{t_{0}}u_{λ},
y_{t_{0}+λ} = y_{t_{0}}v_{λ}.

Notice that ∥u_{λ}∥ = ∥v_{λ}∥ = 1. we claim that {u_{λ} : 0 < λ < T − t_{0}} and {v_{λ} : 0 < λ < T − t_{0}} are left-coherent families. Indeed, if 0 < λ_{1} < λ_{2} < T − t_{0} then since {x_{t}} is left-coherent we may find z ∈ E(λ_{2} −λ_{1}) such that x_{t_{0}+λ_{2}} = x_{t_{0}+λ_{1}}z. Hence

x_{t_{0}}u_{λ_{2}} = x_{t_{0}+λ_{2}} = x_{t_{0}+λ_{1}} = x_{t_{0}}u_{λ_{1}}z.

From the left cancellation property we conclude that u_{λ_{2}} = u_{λ_{1}}z, proving that {u_{λ}} is left-coherent. The same is true of {v_{λ}}.

Thus by Theorem 5.1 we have

lim_{λ→0+}| ⟨x_{t_{0}+λ}, y_{t_{0}+λ}⟩ | = lim_{λ→0+} | ⟨x_{t_{0}}u_{λ}, y_{t_{0}}v_{λ}⟩ | =

| ⟨x_{t_{0}}, y_{t_{0}}⟩ | lim_{λ→0+} | ⟨u_{λ}, v_{λ}⟩ | = | ⟨x_{t_{0}}, y_{t_{0}}⟩ |,

proving right continuity at λ. left continuity. Using left-coherence of {x_{t}} and {y_{t}}, we may find vectors u_{λ}, v_{λ} ∈ E_{λ} for every λ ∈ (0, t_{0}) such that

(5.4)  x_{t_{0}} = x_{t_{0}−λ}u′_{λ}
y_{t_{0}} = y_{t_{0}−λ}v′_{λ}.

Notice that {u′_{λ}} and {v′_{λ}} are right-coherent families. The proof is similar to what was done above. For example, for 0 < λ_{1} < λ_{2} < t_{0} we may find z′ ∈ E(λ_{2} − λ_{1}) such that

x_{t_{0}−λ_{1}} = x_{t_{0}−λ_{2}}z′.

Hence

x_{t_{0}−λ_{2}}u′_{λ_{2}} = x_{t_{0}−λ_{1}}u′_{λ_{1}} = x_{t_{0}−λ_{2}}z′u′_{λ_{1}},

and we obtain

u′_{λ_{2}} = z′u′_{λ_{1}}

after cancelling x_{t_{0}−λ_{2}} from the left. The proof that {v′_{λ}} is right-coherent is of course the same.

Using (5.4) we have

| ⟨x_{t_{0}}, y_{t_{0}}⟩ | = | ⟨x_{t_{0}−λ}, y_{t_{0}−λ}⟩ | · | ⟨u′_{λ}, v′_{λ}⟩ |

for every 0 < λ < t_{0}. Because of Theorem 5.1,

lim_{λ→0+} | ⟨u′_{λ}, v′_{λ}⟩ | = 1,

hence

lim_{λ→0+} | ⟨x_{t_{0}−λ}, y_{t_{0}−λ}⟩ | = | ⟨x_{t_{0}}, y_{t_{0}}⟩ |,

proving left continuity at t_{0}. The continuity of ϕ at t = 0 was established in Theorem 5.1 □

6. Inner products of decomposable vectors.

Let t > 0 and let x ∈ D(t) be a decomposable vector. Then for 0 < s < t there are vectors a_{s}, b_{s} ∈ E(s) such that

(6.1)  x = a_{s}b_{t−s}.

Remark. Note that if ∥x∥ = 1, then ∥a_{s}∥ · ∥b_{t−s}∥ = 1 and we may perform an obvious renormalization to achieve ∥a_{s}∥ = ∥b_{s}∥ = 1 for every s.

It is not obvious that each a_{s} and each b_{s} is a decomposable vector. The following lemma establishes this fact, and moreover it shows that {a_{s} : 0 < s < t} (resp. {b_{s} : 0 < s < t}) is a left-decomposable (resp. right-decomposable) family.

Lemma 6.2. Choose s_{1}, s_{2} with 0 < s_{1} < s_{2} < t, and suppose that a_{i} ∈ E(s_{i}) and b ∈ E(t − s_{i}) satisfy

a_{1}b_{1} = a_{2}b_{2}.

Then a_{1} is a left-divisor of a_{2} and b_{2} is a right-divisor of b_{1}; i.e., there are vectors c_{1}, c_{2} ∈ E(s_{2} − s_{1}) such that

a_{2} = a_{1}c_{1}
b_{1} = c_{2}b_{2}.

proof. We may assume that x = a_{1}b_{1} = a_{2}b_{2} satisfies ∥x∥ = 1, and thus after an obvious renormalization we may also assume ∥a_{i}∥ = ∥b_{i}∥ = 1.

We require an enhanced version of the right cancellation law. Note that if z is any element of E, say z ∈ E(λ) for λ > 0, then the right multiplication map R_{z} : u ∈ E ↦ uz ∈ E restricts to a bounded linear map on fiber spaces, carrying E(μ) to E(μ + λ), and thus has a fiber adjoint. Let

R_{z}^{∗} : {E(μ) : μ > λ} → E

be the total map defined by these adjoints. We claim that that R_{z}^{∗} commutes with any left multiplication operator L_{a} : u ↦ au in the sense that L_{a} commutes with the restriction of R_{z}^{∗} to any fiber space in any case in which the formulas make sense. That is, if a ∈ E(λ) and u ∈ E(μ) with μ > λ, then we have

(6.3)  aR_{z}^{∗}(u) = R_{z}^{∗}(au).

To see this, simply note that since E(μ) is spanned by E(μ − λ)E(λ), it suffices to verify that (6.3) is valid for vectors u of the form u_{1}u_{2} with u_{1} ∈ E(μ − λ) and u_{2} ∈ E(λ). In that case the left side of 6.3 is

aR_{z}^{∗}(u_{1}u_{2}) = a(u_{1} ⟨u_{2}, z⟩)

while the right side is

R_{z}^{∗}(a(u_{1}u_{2})) = R_{z}^{∗}((au_{1})u_{2}) = au_{1} ⟨u_{2}, z⟩ ,

as asserted.

We apply these remarks to the proof of Lemma 6.2 as follows. Write

a_{2} = a_{2}∥b_{2}∥^{2} = R_{b_{2}}^{∗}(a_{2}b_{2}) = R_{b_{2}}^{∗}(a_{1}b_{1}) = a_{1}R_{b_{2}}^{∗}(b_{1}),

and therefore we can take c_{1} = R_{b_{2}}^{∗}(b_{1}) ∈ E(s_{2} − s_{1}).

The other decomposition b_{1} = c_{2}b_{2} follows from this by considering the product system E^{o} opposite to E. Indeed, if we interpret the equation a_{1}b_{1} = a_{2}b_{2} in E^{o}, it becomes b_{1}a_{1} = b_{2}a_{2}. By what was proved above, there is an element c_{2} ∈ E^{o}(s_{2} − s_{1}) for which b_{1} = b_{2}c_{2}, and if we interpret the latter in E we obtain b_{1} = c_{2}b_{2}. □

Remark 6.4. We may conclude that every vector x ∈ D(t) can be associated with a propagator {x(r, s) ∈ D(s − r) : 0 ≤ r < s ≤ t} which satisfies x(0, t) = x. Indeed, for each 0 < s < t we can find a nonzero left-divisor x_{s} ∈ E(s) for x. Set x_{t} = x. By Lemma 6.2, {x_{s} : 0 < s ≤ t} is a left-coherent family with x_{s} ∈ D(s) for every s. Because of the left cancellation law, we can therefore define a propagator {x(r, s) : 0 ≤ r < s ≤ t} by setting

x_{s} = x_{r}x(r, s)

for 0 < r < s ≤ t, and by setting x(0, s) = x_{s}.

Theorem 6.5. For any t > 0 and any two vectors x, y ∈ D(t) we have ⟨x, y⟩ ≠ 0.

proof. There is clearly no loss if we assume that ∥x∥ = ∥y∥ = 1. By the preceding remarks, we can find propagators {x(r, s) : 0 ≤ r < s ≤ t} and {y(r, s) : 0 ≤ r < s ≤ t} for x and y with the property that x(0, t) = x and y(0, t) = y. By an obvious renormalization, we can also assume that ∥x(r, s)∥ = ∥y(r, s)∥ = 1 for every 0 ≤ r < s ≤ t.

Notice that for every 0 < r ≤ t we have

(6.6)  lim_{λ→0+} | ⟨x(r − λ, r), y(r − λ, r)⟩ | = 1.

Indeed, this follows by applying 5.3 to the right-coherent normalized sections a_{λ} = x(r − λ, r) and b_{λ} = y(r − λ, r), 0 < λ < r.

Now the function f : [0, t] → ℝ defined by

f(s) = \{ \begin{matrix}| ⟨x(0, s), y(0, s)⟩ |, & 0 < s ≤ t, \\ 1, & s = 0 \end{matrix}

is continuous, by Theorem 5.3. We claim that f is never zero. For if there were an r ∈ [0, t] for which f(r) = 0, then there is a smallest one r_{0}, and we must have 0 < r_{0} ≤ t. But for every s ∈ (0, r_{0}) we can write

f(r_{0}) =| ⟨x(0, r_{0}), y(0, r_{0})⟩ | = | ⟨x(0, s)x(s, r_{0}), y(0, s)y(s, r_{0})⟩ |
=f(s)| ⟨x(s, r_{0}), y(s, r_{0})⟩ |.

f(s) is nonzero for every s < r_{0}, and because of (6.6) the term | ⟨x(s, r_{0}), y(s, r_{0})⟩ | is nonzero when s is sufficiently close to r_{0}. This contradicts the fact that f(r_{0}) was supposed to be zero. Taking s = t we find that | ⟨x, y⟩ | = f(t) ≠ 0 □

7. Continuity and normalization of the inner product.

We will write D for the set of all left-coherent decomposable sections. Thus D consists of all sections

t ∈ (0, ∞) ↦ x_{t} ∈ E(t)

which are left-coherent and for which x_{t} is never zero. It is possible, of course, that D = ∅. But if D is not empty then we are interested in establishing continuity of the inner product function

(7.1)  t ∈ (0, ∞) ↦ ⟨x_{t}, y_{t}⟩ ,

defined by two elements x, y ∈ D. In this section we will show that if one normalizes the elements of D appropriately then inner products of the form (7.1) are continuous.

Remarks. Note that in general, nothing can be said about continuity of the inner products (7.1) (see the remarks following Theorem 5.1).

Notice too that, even though unique factorization fails in the multiplicative family {D(t) : t > 0} we still have a left cancellation law, and this implies that there is a bijective correspondence between left-coherent sections and propagators. For example, if x ∈ D(t) and we find a left coherent family {x_{s} : 0 < s ≤ t} for which x_{t} = x then we may define a unique propagator {x(r, s) ∈ D(t − s) : 0 ≤ r < s ≤ t} by

x_{s} = x_{r}x(r, s)

for 0 < r < s (by the left cancellation property), and where for r = 0 we put x(0, s) = x_{s}. The left-coherent family is recovered from its propagator via x_{s} = x(0, s), 0 < s ≤ t.

The normalization in D is done as follows. Choose an arbitrary element e ∈ D satisfying ∥e_{t}∥ = 1 for every t > 0; e will be fixed throughout the remainder of this section.

Definition 7.3. D^{e} is the set of all x ∈ D satisfying ⟨x_{t}, e_{t}⟩ = 1 for every t > 0.

Lemma 7.4. For every x ∈ D^{e}, the norm ∥x_{t}∥ is a continuous nondecreasing function of t with

lim_{t→0+} ∥x_{t}∥ = 1.

In particular, we have ∥x_{t}∥ ≥ 1 for every t > 0.

proof. Let u_{t} = ∥x_{t}∥^{−1}x_{t}. Then both {e_{t}} and {u_{t}} are left-coherent families of unit vectors. So by Theorems 5.1 and 5.3 we may conclude that | ⟨u_{t}, e_{t}⟩ | is continuous in t over the interval (0, ∞) and satisfies

lim_{t→0} | ⟨u_{t}, e_{t}⟩ | = 1.

Since

| ⟨u_{t}, e_{t}⟩ | = ∥x_{t}∥^{−1}| ⟨x_{t}, e_{t}⟩ | = ∥x_{t}∥^{−1},

the continuity assertion follows.

To see that ∥x_{t}∥ increases with t, choose 0 < s < t. By left-coherence of {x_{t}} and {e_{t}} we can write

e_{t} = e_{s}u,
x_{t} = x_{s}v,

where u = e(s, t), and v = x(s, t) belong to E(t − s). Note that u must be a unit vector because ∥e(s, t)∥ = ∥e_{t}∥/∥e_{s}∥ = 1. Notice too that ∥v∥ ≥ 1. Indeed, since ⟨e_{r}, x_{r}⟩ = 1 for all r we can write

⟨u, v⟩ = ⟨e_{s}, x_{s}⟩ ⟨u, v⟩ = ⟨e_{s}u, x_{s}v⟩ = ⟨e_{t}, x_{t}⟩ = 1,

so by the Schwarz inequality

1 = | ⟨u, v⟩ | ≤ ∥u∥ · ∥v∥ = ∥v∥.

It follows that

∥x_{s}∥ ≤ ∥x_{s}∥ · ∥v∥ = ∥x_{s}v∥ = ∥x_{t}∥

as asserted □

Our principal result on the continuity of inner products is the following.

Theorem 7.5. Let x, y ∈ D^{e}. Then the inner product ⟨x_{t}, y_{t}⟩ is continuous and nonzero on 0 < t < ∞, and satisfies

lim_{t→0+} ⟨x_{t}, y_{t}⟩ = 1.

proof. We will deduce Theorem 7.5 from the following inequality. For every s, t, T satisfying 0 < s < t ≤ T < ∞ we claim that

(7.6)  | ⟨x_{s}, y_{s}⟩ − ⟨x_{t}, y_{t}⟩ | ≤ ∥x_{T} ∥ · ∥y_{T} ∥\sqrt{(∥x_{t}∥^{2} − ∥x_{s}∥^{2})(∥y_{t}∥^{2} − ∥y_{s}∥^{2})}.

To prove (7.6), we write

x_{t} = x_{s}u
y_{t} = y_{s}v
e_{t} = e_{s}f

where u = x(s, t), v = y(s, t), f = e(s, t) ∈ E(t − s) and ∥f∥ = 1. Notice that

(7.7)  ⟨u, f⟩ = ⟨v, f⟩ = 1.

Indeed, since ⟨x_{s}, e_{s}⟩ = 1 we have

⟨u, f⟩ = ⟨x_{s}, e_{s}⟩ ⟨u, f⟩ = ⟨x_{s}u, e_{s}f⟩ = ⟨x_{t}, e_{t}⟩ = 1,

and similarly ⟨v, f⟩ = 1. We can therefore estimate the quantity ⟨u, v⟩−1 as follows:

| ⟨u, v⟩ − 1| = | ⟨u − f, v − f⟩ | ≤ ∥u − f∥ · ∥v − f∥.

By Lemma 7.3 we have ∥x_{s}∥ ≥ 1. Hence we may use ⟨u, f⟩ = 1 again to obtain

∥u − f∥^{2} = ∥u∥^{2} − 1 = \frac{∥x_{t}∥^{2}}{∥x_{s}∥^{2}} − 1 = ∥x_{s}∥^{−2}(∥x_{t}∥^{2} − ∥x_{s}∥^{2}) ≤ ∥x_{t}∥^{2} − ∥x_{s}∥^{2}.

Similarly,

∥v − f∥ ≤ \sqrt{∥y_{s}∥^{2} − ∥y_{s}∥^{2}}.

Thus

| ⟨u, v⟩ − 1| ≤ \sqrt{(∥x_{t}∥^{2} − ∥x_{s}∥^{2})(∥y_{t}∥^{2} − ∥y_{s}∥^{2})}.

The inequality (7.6) follows after multiplying the preceding inequality through by | ⟨x_{s}, y_{s}⟩ |, noting that

| ⟨x_{s}, y_{s}⟩ | · | ⟨u, v⟩ − 1| = | ⟨x_{s}, y_{s}⟩ ⟨u, v⟩ − ⟨x_{s}, y_{s}⟩ |
= | ⟨x_{t}, y_{t}⟩ − ⟨x_{s}, y_{s}⟩ |,

and using the Schwarzenegger inequality and Lemma 7.4 to estimate the factor | ⟨x_{s}, y_{s}⟩ | on the right by way of

| ⟨x_{s}, y_{s}⟩ | ≤ ∥x_{s}∥ · ∥y_{s}∥ ≤ ∥x_{T} ∥ · ∥y_{T} ∥.

This establishes (7.6).

Now Lemma 7.4 implies that ∥x_{t}∥^{2} and ∥y_{t}∥^{2} are continuous increasing functions tending to 1 as t → 0+, and from (7.6) we immediately conclude that ⟨x_{t}, y_{t}⟩ is continuous on (0, ∞). If we allow s to tend to 0 in (7.6) and use lim_{t→0+} ∥x_{s}∥ = 1 from Lemma 7.4 the result is

| ⟨x_{t}, y_{t}⟩ − 1| ≤ ∥x_{T} ∥ · ∥y_{T} ∥\sqrt{(∥x_{t}∥^{2} − 1)(∥y_{t}∥^{2} − 1)},

from which we deduce

lim_{t→0+} ⟨x_{t}, y_{t}⟩ = 1.

That establishes continuity on the closed interval [0, ∞) □

8. Continuous logarithms.

Fix t > 0. We have seen above that if x, y are two decomposable vectors in E(t) then the inner product ⟨x, y⟩ is not zero. Thus one might attempt to define a logarithm function (x, y) ∈ D(t) ↦ L(t; x, y) ∈ ℂ with the property that

e^{L(t;x,y)} = ⟨x, y⟩ ,

in such a way that the logarithms fit together consistently for different values of t. We will show that this is in fact possible, provided that one is careful to define the logarithm so as to remove as much ambiguity as possible. In section 9 we will show that L is conditionally positive definite.

Let Δ = {Δ(t) : t > 0} be the path space obtained from {D(t) : t > 0} as in the introduction to Part II above. Despite the fact that Δ and Δ^{2} are lifeless sets, there is a useful notion of continuity for complex functions defined on them. We will say that ϕ : Δ → ℂ is continuous if, for every left-coherent section t ∈ (0, ∞) ↦ u_{t} ∈ D(t), the function f(t) = ϕ(t; u_{t}) is continuous on the interval (0, ∞), and the limit

f(0) = lim_{t→0+} f(t)

exists. For functions ψ : Δ^{2} → ℂ, continuity means that for any pair of sections u, v ∈ D, the function g(t) = ψ(t; u_{t}, v_{t}) is continuous for positive t and extends continuously to [0, ∞). We will say that ϕ (resp. ψ) vanishes at the origin if f(0) = 0 (resp. g(t) = 0) for all choices of u (resp. u, v).

Remark 8.1. Notice that certain normalized inner products give rise to continuous functions F : Δ^{2} → ℂ. For example, with e as above put

F (t; x, y) = \frac{⟨x, y⟩}{⟨x, e_{t}⟩ ⟨e_{t}, y⟩},

for t > 0, x, y ∈ D(t). To see that F is continuous choose u, v ∈ D and put u′_{t} = ⟨u_{t}, e_{t}⟩^{−1} u_{t} and v′_{t} = ⟨v_{t}, e_{t}⟩^{−1} v_{t}. Then we have

F (t; u_{t}, v_{t}) = ⟨u′_{t}, v′_{t}⟩

because of the homogeniety of F. Theorem 7.5 implies that the right side is continuous in t and tends to 1 as t → 0+. It follows that F : Δ^{2} → ℂ is a continuous function. Needless to say, F depends on e.

Theorem 8.2. Let e ∈ D satisfy ∥e_{t}∥ = 1 for every t > 0. Then there is a unique continuous function L^{e} : Δ^{2} → ℂ which vanishes at the origin and satisfies

(8.2a)  e^{L^{e}(t;x,y)} = \frac{⟨x, y⟩}{⟨x, e_{t}⟩ ⟨e_{t}, y⟩},

for every t > 0, x, y ∈ D(t).

If f ∈ D satisfies ∥f_{t}∥ = 1 for every t > 0 and L^{f} : Δ^{2} → ℂ is the corresponding logarithm, then there is a continuous function ϕ : Δ → ℂ which vanishes at 0 and satisfies

(8.2b)  L^{f}(t; x, y) = L^{e}(t; x, y) + ϕ(t; x) + \overline{ϕ(t; y)}

for all t > 0, x, y ∈ D(t).

Remark 8.3. The function L^{e} : Δ^{2} → ℂ is called the e-logarithm of the inner product on E. Notice that we can use L^{e} to define a logarithm of the non-normalized inner product in the following way. For every t > 0 and every x ∈ D(t) let f(t; x) be a complex number such that

e^{f(t;x)} = ⟨x, e_{t}⟩ .

The function f need have no regularity properties whatsoever, and may even be non-measurable. Nevertheless, once we settle on f then we can define

L(t; x, y) = L^{e}(t; x, y) + f(t; x) + \overline{f}(t; y)

and this new L will satisfy

e^{L(t;x,y)} = ⟨x, y⟩ , x, y ∈ D(t), t > 0.

We will see below that L^{e}(t; ·, ·) is a positive definite function on D(t) × D(t), and hence L(t; ·, ·) is a conditionally positive definite logarithm of the inner product ⟨·, ·⟩ : D(t)×D(t) → ℂ. Now the construction of a Hilbert space from a conditionally positive definite function has the property that L(t; ·, ·) and L^{e}(t; ·, ·) determine the same Hilbert space (see section 7). It follows that the function f has no effect on the invariantly defined Hilbert spaces that concern us. Moreover, for the same reason (8.2b) implies that these Hilbert spaces will also be independent of the particular choice of normalized section e ∈ D.

proof of Theorem 8.2. For uniqueness, notice that if L^{e}, M^{e} : Δ^{2} → ℂ both satisfy the conditions associated with (8.2a), then

ϕ(t; x, y) = L^{e}(t; x, y) − M^{e}(t; x, y)

is a continuous complex-valued function on Δ^{2} which vanishes at the origin and satisfies

e^{ϕ(t;x,y)} = 1

identically. To see that ϕ = 0 choose t > 0 and x, y ∈ D(t). Let u, v ∈ D be leftcoherent sections such that u_{t} and v_{t} are, respectively, scalar multiples of x and y (see Theorem 10.1). The function s ∈ (0.∞) → ϕ(t; u_{s}, v_{s}) ∈ ℂ is continuous, vanishes as s → 0+ and satisfies

e^{ϕ(s;u_{s},v_{s})} = 1

for all s > 0. Hence ϕ(s; u_{s}, v_{s}) = 0 for all s. By homogeneity, ϕ(t; x, y) = ϕ(t; u_{t}, v_{t}) = 0.

For existence, fix t > 0, x, y ∈ D(t). We define L^{e}(t; x, y) as follows. Again, we find sections u, v ∈ D such that u_{t} = λx, v_{t} = μy, with λμ ≠ 0. Now the function

s ∈ (0, ∞) ↦ \frac{⟨u_{s}, v_{s}⟩}{⟨u_{s}, e_{s}⟩ ⟨e_{s}, v_{s}⟩}

is continuous, never 0, and tends to 1 as s → 0+ (see Remark 8.1). Thus there is a unique continuous function l ∈ C[0, ∞) satisfying the conditions l(0) = 0 and

(8.4)  e^{l(s)} = \frac{⟨u_{s}, v_{s}⟩}{⟨u_{s}, e_{s}⟩ ⟨e_{s}, v_{s}⟩},  s > 0.

We define L^{e}(t; x, y) = l(t).

To see that L^{e}(t; x, y) is well-defined, choose another pair u′, v′ ∈ D so that u′_{t} = λ′x and v′_{t} = μ′y with λ′μ′ ≠ 0. Choose l′ ∈ C[0, ∞) with l′(0) = 0 so that (8.4) is satisfied with u′, v′ replacing u, v. We have to show that l′(t) = l(t). But for 0 < s ≤ t the uniqueness of factorizations of the two vectors x, y implies that there are nonzero complex numbers α_{s}, β_{s} so that u′_{s} = α_{s}u_{s}, v′_{s} = β_{s}v_{s} for 0 < s ≤ t. It follows that the right side of (8.4) is unaffected by passing from u, v to u′, v′. Hence l′ = l and finally l′(t) = l(t).

Notice that the continuity of L^{e} follows from its definition. Indeed, for u, v and l related by (8.4) in the definition of L^{e}(t; x, y), we must also have

L^{e}(s; u_{s}, v_{s}) = l(s)

for every 0 < s ≤ t. In particular, the function s ∈ (0, t] ↦ L^{e}(s; u_{s}, v_{s}) is continuous and tends to 0 as s → 0+. Since t, x, y can be chosen arbitrarily, the sections u, v ∈ D are also arbitrary. It follows that L^{e} : Δ^{2} → ℂ is continuous and vanishes at 0.

To prove (8.2b), pick f ∈ D so that ∥f_{t}∥ = 1 for every t, and consider the function Φ : Δ → ℂ defined by

Φ(t; x) = \frac{| ⟨e_{t}, f_{t}⟩ | ⟨x, e_{t}⟩}{⟨f_{t}, e_{t}⟩ ⟨x, f_{t}⟩}, t > 0, x, y ∈ D(t).

We claim that for every u ∈ D, Φ(t, u_{t}) is continuous in t ∈ (0, ∞) and tends to 1 as t → 0+. Indeed, putting u′ = ⟨u_{t}, f_{t}⟩^{−1} u_{t} and e′_{t} = ⟨e_{t}, f_{t}⟩^{−1} e_{t}, then e′ and u′ are elements of D satisfying ⟨e′_{t}, f_{t}⟩ = ⟨u′_{t}, f_{t}⟩ = 1, and

Φ(t; u_{t}) = | ⟨e_{t}, f_{t}⟩ | ⟨u′_{t}, e′_{t}⟩

for t > 0. The claim follows because both | ⟨e_{t}, f_{t}⟩ | and ⟨u′_{t}, e′_{t}⟩ are continuous in t and tend to 1 as t → 0+ by Theorem 7.5.

To define ϕ we proceed as we did in defining the function L^{e} above. Fix t > 0, x ∈ D(t), and choose u ∈ D so that u_{t} = λx for some complex number λ ≠ 0. By the preceding paragraph there is a unique continuous function l ∈ C[0, t] so that l(0) = 0 and

e^{l(s)} = Φ(s; u_{s}), 0 < s ≤ t.

Put ϕ(t; x) = l(t). One shows that ϕ is well-defined and continuous as one did for L^{e}.

Finally, since both L^{f}(t; x, y) − L^{e}(t; x, y) and ϕ(t; x) + \overline{ϕ(t; y)} define continuous functions on Δ^{2} which vanish at 0, (8.2b) will follow if we show that

(8.5)  e^{L^{f} (t;x,y)−L^{e}(t;x,y)} = e^{ϕ(t;x)+\overline{ϕ(t;y)}}.

The left side of (8.5) is the quotient of

e^{L^{f} (t;x,y)} = \frac{⟨x, y⟩}{⟨x, f_{t}⟩ ⟨f_{t}, y⟩}

by the quantity

e^{L^{e}(t;x,y)} = \frac{⟨x, y⟩}{⟨x, e_{t}⟩ ⟨e_{t}, y⟩}.

Thus the ⟨x, y⟩ terms cancel out and the left side of (8.5) reduces to

(8.6)  \begin{matrix}⟨x, e_{t}⟩ & ⟨e_{t}, y⟩ \\ ⟨x, f_{t}⟩ & ⟨f_{t}, y⟩ \end{matrix}.

Similarly, the right side is the product of

e^{ϕ(t;x)} = \frac{| ⟨e_{t}, f_{t}⟩ | ⟨x, e_{t}⟩}{⟨f_{t}, e_{t}⟩ ⟨x_{t}, f_{t}⟩}

with the quantity

\overline{e^{ϕ(t;y)}} = \frac{| ⟨e_{t}, f_{t}⟩ | ⟨e_{t}, y⟩}{⟨e_{t}, f_{t}⟩ ⟨f_{t}, y⟩}.

After performing the indicated multiplication all terms involving ⟨e_{t}, f_{t}⟩ cancel and the result agrees with (8.6) □

Remark 8.7. We remark that for each e, the e-logarithm is self-adjoint in the sense that

\overline{L^{e}(t; x, y)} = L^{e}(t; y, x), t > 0, x, y ∈ D(t).

To see that, simply note that the function F : Δ^{2} → ℂ defined by

F (t; x, y) = \overline{L^{e}(t; y, x)}

has all of the defining properties of an e-logarithm, and hence F = L^{e} by uniqueness.

In the applications of part III, we will need to know that the function L^{e} defines an addtitive form on Δ^{2}. The following establishes this fact.

Proposition 8.8. Fix s > 0. Then there is a continuous function ψ_{s} : Δ → ℂ, vanishing at 0, such that for all x_{1}, x_{2} ∈ Δ(s), all t > 0 and all y_{1}, y_{2} ∈ Δ(t) we have

(8.9) L^{e}(s + t; x_{1}y_{1}, x_{2}y_{2}) − L^{e}(s; x_{1}, x_{2}) − L^{e}(t; y_{1}, y_{2}) = ψ_{s}(t; y_{1}) + \overline{ψ_{s}(t; y_{2})}.

proof. Fix s > 0. We claim that there is a continuous function ψ_{s} : Δ → ℂ which vanishes at 0 and satisfies

e^{ψ_{s}(t;y)} = \frac{| ⟨e(s, s + t), e_{t}⟩ | ⟨y, e_{t}⟩}{⟨y, e(s, s + t)⟩ ⟨e(s, s + t), y⟩}

for all y ∈ D(t), t > 0. In order to define ψ_{s}, fix t > 0 and y_{0} ∈ D(t). Choose a left-decomposable section y ∈ D such that y_{t} is a scalar multiple of y_{0}. The function

t ∈ (0, ∞) ↦ | ⟨e(s, s + t), e_{t}⟩ |

is continuous and tends to 1 as t → 0+ by Theorem 2.3. Similarly, by Remark 6.1, the function

t ∈ (0, ∞) → \frac{⟨y_{t}, e_{t}⟩}{⟨y_{t}, e(s, s + t)⟩ ⟨e(s, s + t), y_{t}⟩}

has the same properties. Thus

t ∈ (0, ∞) → \frac{| ⟨e(s, s + t), e_{t}⟩ | ⟨y_{t}, e_{t}⟩}{⟨y_{t}, e(s, s + t)⟩ ⟨e(s, s + t), y_{t}⟩}

is a continuous function which tends to 1 as t → 0+. It follows that there is a unique continuous function l : [0, ∞) → ℂ such that l(0) = 0 and

e^{l(t)} = \frac{| ⟨e(s, s + t), e_{t}⟩ | ⟨y_{t}, e_{t}⟩}{⟨y_{t}, e(s, s + t)⟩ ⟨e(s, s + t), y_{t}⟩}

for t > 0. We define ψ_{s}(t; y_{0}) = l(t).

ψ_{s}(t; ·) is a homogeneous function of degree 0 on D(t) and hence we may consider ψ_{s} to be a function defined on Δ. ψ_{s} is continuous because of the way it was defined.

It remains to show that ψ_{s} satisfies (8.9). For that, it suffices to show that for any pair of left-decomposable sections t → y_{t}, y′_{t} ∈ D(t) we have

L^{e}(s + t; x_{1}y_{t}, x_{2}y′_{t}) − L^{e}(s; x_{1}, x_{2}) − L^{e}(t; y_{t}, y′_{t}) = ψ_{s}(t; y_{t}) + \overline{ψ_{s}(t; y′_{t})}

for every t > 0. To see this, let L(t) and R(t) be the left and right sides of the preceding formula. R(t) is continuous on (0, ∞) and tend to 0 as t → 0+ by definition of ψ_{s}, and we claim that L(·) has these two properties as well. Indeed, L(t) is continuous for positive t because of the continuity of L^{e}. To see that L(t) → 0 as t → 0+, consider the left-coherent sections u, u′ defined by

u_{r} = \{ \begin{matrix}(x_{1})_{r}, & 0 < r ≤ s \\ (x_{1})_{s}y_{r−s}, & r > s \end{matrix} , u′_{r} = \{ \begin{matrix}(x_{2})_{r}, & 0 < r ≤ s \\ (x_{2})_{s}y′_{r−s}, & r > s \end{matrix} .

By continuity of L^{e}(r; u_{r}, u′_{r}) at r = s we obtain

lim_{t→0+} L(t) = lim_{t→0+}(L^{e}(s + t; u_{s+t}, u′_{s+t}) − L^{e}(s; x_{1}, x_{2}) − L^{e}(t; y_{t}, y′_{t}))
= L^{e}(s; u_{s}, u′_{s}) − L^{e}(s; x_{1}, x_{2}) = 0,

as asserted.

Since both L(t) and R(t) are continuous on (0, ∞) and tend to 0 as t → 0+, it suffices to show that

e^{L(t)} = e^{R(t)}, for t > 0.

But

e^{R(t)} =\frac{| ⟨e(s, s + t), e_{t}⟩ | ⟨y_{t}, e_{t}⟩}{⟨y_{t}, e(s, s + t)⟩ ⟨e(s, s + t), e_{t}⟩} · (\overline{\frac{| ⟨e(s, s + t), e_{t}⟩ | ⟨y′_{t}, e_{t}⟩}{⟨y′_{t}, e(s, s + t)⟩ ⟨e(s, s + t), e_{t}⟩}})

=\begin{matrix}⟨y_{t}, e_{t}⟩ & ⟨e_{t}, y′_{t}⟩ \\ ⟨y_{t}, e(s, s + t)⟩ & ⟨e(s, s + t), y′_{t}⟩ \end{matrix}

while

e^{L(t)} = \frac{⟨x_{1}, y_{t}, x_{2}y′_{t}⟩}{⟨x_{1}y_{t}, e_{s+t}⟩ ⟨e_{s+t}, x_{2}y′_{t}⟩} · \frac{⟨x_{1}, e_{s}⟩ ⟨e_{s}, y_{2}⟩}{⟨x_{1}, x_{2}⟩} · \frac{⟨y_{1}, e_{t}⟩ ⟨e_{t}, y_{2}⟩}{⟨y_{1}, y_{2}⟩}.

Using the formulas

⟨x_{1}y_{t}, x_{2}y′_{t}⟩ = ⟨x_{1}, x_{2}⟩ ⟨y_{t}, y′_{t}⟩
⟨x_{1}y_{t}, e_{s+t}⟩ = ⟨x_{1}y_{t}, e_{s}e(s, s + t)⟩ = ⟨x_{1}, e_{s}⟩ ⟨y_{t}, e(s, s + t)⟩
⟨e_{s+t}, x_{2}y′_{t}⟩ = ⟨e_{s}, x_{2}⟩ ⟨e(s, s + t), y′_{t}⟩

and performing the obvious cancellations, we obtain

e^{L(t)} = \begin{matrix}⟨y_{t}, e_{t}⟩ & ⟨e_{t}, y′_{t}⟩ \\ ⟨y_{t}, e(s, s + t)⟩ & ⟨e(s, s + t, y′_{t}⟩ \end{matrix} = e^{R(t)},

as required □

9. Infinite divisibility of the inner product.

We have indicated in remark 8.3 how to find functions L of the form

L(t; x, y) = L^{e}(t; x, y) + f(t, x) + \overline{f(t; y)}

that are logarithms of the inner product restricted to decomposable vectors:

e^{L(t;x,y)} = ⟨x, y⟩ , t > 0, x, y ∈ D(t).

It is essential for the constructions of Part III that such an L should have the property that for fixed t > 0, it defines a conditionally positive definite function on D(t) × D(t). According to the remarks at the beginning of section 2, it would be enough to exhibit a sequence of positive definite functions Φ_{n} : D(t) × D(t) → ℂ (depending on t) such that

Φ_{n}(x, y)^{n} = ⟨x, y⟩ , x, y ∈ D(t)

for every n = 1, 2, . . .. Unfortunately, there are no natural candidates for the positive definite functions Φ_{n}. Thus we will have to establish the conditional positive definiteness of L(t; ·, ·) directly, by making use of the structure of the product system itself. Actually, we will prove somewhat more than we require.

Theorem 9.2. Let e ∈ D satisfy ∥e_{t}∥ = 1, t > 0. Then for every t > 0 the function

(x, y) ∈ D(t) × D(t) ↦ L^{e}(t; x, y)

is positive definite.

The proof of Theorem 9.2 will occupy the remainder of this section. Note that once 9.2 has been proved, one can immediately deduce Corollary 9.3. For every t > 0, the inner product of E restricts to an infinitely divisible positive definite function on D(t) × D(t).

proof of Theorem 9.2. Let e be an element of D satisfying ∥e_{t}∥ = 1 for all t > 0, which will be fixed throughout the remainder of this section. We will define a function

P^{e} : Δ^{2} → ℂ

with the property that each function

(x, y) ∈ D(t) × D(t) ↦ P^{e}(t; x, y)

is obviously positive definite, and which is also a continuous e-logarithm that vanishes 0. The conclusion L^{e} = P^{e} will then follow by the uniqueness assertion of Theorem 8.2 and hence we obtain 9.2.

Turning now to the proof, fix t > 0 and choose x, y ∈ D(t) with ⟨x, e_{t}⟩ = ⟨y, e_{t}⟩ = 1. Because of the normalization of x and y there are left-coherent families {x_{s} : 0 < s ≤ t} and {y_{s} : 0 < s ≤ t} satisfying ⟨x_{s}, e_{s}⟩ = ⟨y_{s}, e_{s}⟩ = 1 for all s with the property that x_{t} = x and y_{t} = y. Moreover, the two families are uniquely determined by these conditions because of the uniqueness of factorizations. If I = (a, b] is a subinterval of (0, t] having positive length then we will write x_{I} (resp. y_{I}) for the value of the propagator x(a, b) (resp. y(a, b)). Finally, if P = {0 = s_{0} < s_{1} < ⋯ < s_{m} = t} and Q = {0 = t_{0} < t_{1} < ⋯ < t_{n} = t} are two partitions of (0, t], we will write P ≤ Q for the usual ordering P ⊆ Q. Thus we can define a net of complex numbers on the increasing directed set of partitions by

(9.4)  B_{P}(t; x, y) = ∑_{I∈P}(⟨x_{I}, y_{I}⟩ − 1).

Proposition 9.5. For each t > 0 and every partition P of (0, t], B_{P}(t; ·, ·) is a positive definite function whose associated Hilbert space is separable. The net is decreasing in the sense that for P ≤ Q, B_{P}(t; ·, ·) − B_{Q}(t; ·, ·) is a positive definite function on D(t) × D(t).

proof. Let P = {0 = t_{0} < t_{1} < ⋯ < t_{n} = t}. To see that B_{P}(t; ·, ·) is positive definite, choose x, y ∈ D(t) and let {x_{s} : 0 < s ≤ t} and {y_{s} : 0 < s ≤ t} be the unique families defined above. It will be convenient to write x_{k} = x(t_{k−1}, t_{k}), y_{k} = y(t_{k−1}, t_{k}) and e_{k} = e(t_{k−1}, t_{k}). Noting that ⟨x_{k}, e_{k}⟩ = ⟨y_{k}, e_{k}⟩ = 1 we have ⟨x_{k}, y_{k}⟩ − 1 = ⟨x_{k} − e_{k}, y_{k} − e_{k}⟩ and thus we can write

B_{P}(t; x, y) = ∑_{k=1}^{n}(⟨x_{k}, y_{k}⟩ − 1) = ∑_{k=1}^{n} ⟨x_{k} − e_{k}, y_{k} − e_{k}⟩ .

Notice that x_{k} − e_{k} and y_{k} − e_{k} belong to the Hilbert space E(t_{k} − t_{k−1}) for every k = 1, 2, …, n. Thus we can define a vector-valued function

F_{P} : D(t) → E(t_{1}) ⊕ E(t_{2} − t_{1}) ⊕ ⋯ ⊕ E(t_{n} − t_{n−1})

by way of

F_{P}(x) = (x_{1} − e_{1}, x_{2} − e_{2}, …, x_{n} − e_{n}).

The preceding formula for B_{P}(t; x, y) now becomes

B_{P}(t; x, y) = ⟨F_{P}(x), F_{P}(y)⟩ .

This formula shows that B_{P}(t; ·, ·) is a positive definite function. Moreover, since the range of the function F_{P} is contained in a separable Hilbert space, it follows that the Hilbert space associated with B_{P}(t; ·, ·) is separable.

It remains to show that if P_{1} and P_{2} are two partitions satisfying P_{1} ≤ P_{2} then B_{P_{1}}(t; ·, ·) − B_{P_{2}}(t; ·, ·) is a positive definite function. Now since the partial order of positive definite functions defined by (B_{1} ≤ B_{2} ⟺ B_{2} − B_{1} is positive definite) is transitive and since P_{2} is obtained from P_{1} by a sequence of steps in which one refines a single interval at every step, we can reduce to the case in which P_{1} = {0 = s_{0} < s_{1} < ⋯ < s_{m} = t} and P_{2} is obtained by adding a single point c to P_{1}, where s_{k−1} < c < s_{k} for some k = 1, 2, …, m. In this case the difference Δ = B_{P_{1}} − B_{P_{2}} is given by

Δ(x, y) = ⟨x(s_{k−1}, s_{k}), y(s_{k−1}, s_{k})⟩ − 1
− (⟨x(s_{k−1}, c), y(s_{k−1}, c)⟩ + ⟨x(c, s_{k}), y(c, s_{k})⟩ − 2)
(9.6)  = ⟨x(s_{k−1}, s_{k}), y(s_{k−1}, s_{k})⟩ − ⟨x(s_{k−1}, c), y(s_{k−1}, c)⟩ − ⟨x(c, s_{k}), y(c, s_{k})⟩
+ 1.

If we write x_{1} = x(s_{k−1}, c), x_{2} = x(c, s_{k}), y_{1} = y(s_{k−1}, c), y_{2} = y(c, s_{k}) then the right side of (9.6) can be rewritten as follows

⟨x_{1}x_{2}, y_{1}y_{2}⟩ − ⟨x_{1}, y_{1}⟩ − ⟨x_{2}, y_{2}⟩ + 1
= ⟨x_{1}, y_{1}⟩ ⟨x_{2}, y_{2}⟩ − ⟨x_{1}, y_{1}⟩ − ⟨x_{2}, y_{2}⟩ + 1

= (⟨x_{1}, y_{1}⟩ − 1)(⟨x_{2}, y_{2}⟩ − 1) = ⟨x_{1} − e_{1}, y_{1} − e_{1}⟩ ⟨x_{2} − e_{2}, y_{2} − e_{2}⟩

= ⟨(x_{1} − e_{1})(x_{2} − e_{2}), (y_{1} − e_{1})(y_{2} − e_{2})⟩

where e_{1} = e(s_{k−1}, c), e_{2} = e(c, s_{k}), and where the inner product in the last term on the right is taken in the Hilbert space E((s_{k} − c) + (c − s_{k−1}) = E(s_{k} − s_{k−1}). The last term clearly defines a positive definite function of x and y □

The conditions of Proposition 9.5 imply that the pointwise limit lim_{P} B_{P}(t; x, y) exists. This is a consequence of the following elementary result.

Lemma 9.7. Let I be a directed set and let {P_{α} : α ∈ I} be a net of positive definite functions on a set X which is decreasing in the sense that α ≤ β implies that P_{α} − P_{β} is positive definite. Then

P_{∞}(x, y) = lim_{α} P_{α}(x, y)

exists for every x, y ∈ X and P_{∞} is a positive definite function. If the Hilbert space associated with some P_{α} is separable then so is the Hilbert space associated with P_{∞}.

proof. Since a pointwise limit of positive definite functions is obviously positive definite, we merely show that the above limit exists and defines a separable Hilbert space.

Fix two elements x, y ∈ X. For every α ∈ I consider the 2 × 2 complex matrix

A_{α} = ( \begin{matrix}P_{α}(x, x) & P_{α}(x, y) \\ P_{α}(y, x) & P_{α}(y, y) \end{matrix} ) .

We may consider {A_{α} : α ∈ D} as a net of self adjoint operators on the two dimensional Hilbert space ℂ^{2}. We have A_{α} ≥ 0 for every α because P_{α} is positive definite, and α ≤ β ⟹ A_{β} ≤ A_{α} because the net P_{α} is decreasing. Hence the net of operators A_{α} must converge in the weak operator topology to a positive operator

A = lim_{α} A_{α}.

Considering A as a 2 × 2 matrix, the element λ in the 12 position satisfies

λ = lim_{α} P_{α}(x, y),

establishing the existence of the required limit.

For separability, notice that there are Hilbert spaces H_{α}, H_{∞} and functions F_{α} : X → H_{α}, F_{∞} : X → H_{∞} with the property

P_{α}(x, y) = ⟨F_{α}(x), F_{α}(y)⟩ ,
P_{∞}(x, y) = ⟨F_{∞}(x), F_{∞}(y)⟩ ,

and where we may also assume H_{α} (resp. H_{∞}) is spanned by F_{α}(X) (resp. F_{∞}(X)). By hypothesis, we can find α so that H_{α} is separable. Since P_{α} − P_{∞} is positive definite it follows that there is a unique contraction T : H_{α} → H_{∞} having the property T (F_{α}(x)) = F_{∞}(x) for every x ∈ X. Thus T maps H_{α} onto a dense subspace of H_{∞}. Since H_{α} is separable we conclude that H_{∞} is separable as well. □

By 9.5 and 9.7, we may define a positive definite function P_{∞}(t; ·, ·) on D(t)×D(t) by

B_{∞}(t; x, y) = lim_{P} B_{P}(t; x, y).

Finally, we define P^{e} : Δ^{2} → ℂ by

P^{e}(t; x, y) = B_{∞}(t; ⟨x, e_{t}⟩^{−1} x, ⟨y, e_{t}⟩^{−1} y).

It remains to show that P^{e} is an e-logarithm, i.e., that it is continuous, vanishes at the origin, and exponentiates correctly.

We deal first with continuity and vanishing at 0. Choose u, v ∈ D so that ⟨u_{t}, e_{t}⟩ = ⟨v_{t}, e_{t}⟩ = 1 for every t. We have to show that the function

t ∈ (0, ∞) ↦ P^{e}(t; u_{t}, v_{t}) = B_{∞}(t; u_{t}, v_{t})

is continuous on (0, ∞) and tends to 0 as t → 0+. This will follow from the following two estimates.

Proposition 9.8. If 0 < s < t and u, v are as above, then

|P^{e}(s; u_{s}, v_{s})| ≤(∥u_{s}∥^{2} − 1)(∥v_{s}∥^{2} − 1),
|P^{e}(t; u_{t}, v_{t}) − P^{e}(s; u_{s}, v_{s})| ≤(∥u_{t}∥^{2} − ∥u_{s}∥^{2})(∥v_{t}∥^{2} − ∥v_{s}∥^{2}).

proof. Consider the first of the two inequalities. Because of the fact that

P^{e}(s; u_{s}, v_{s}) = lim_{P} B_{P}(s; u_{s}, v_{s}),

it suffices to show that for every partition P = {0 = s_{0} < s_{1} < ⋯ < s_{m} = s} of the interval (0, s] we have

(9.9)  |B_{P}(s; u_{s}, v_{s})|^{2} ≤ (∥u_{s}∥^{2} − 1)(∥v_{s}∥^{2} − 1).

For that, let us write u^{k} = u(s_{k−1}, s_{k}), v^{k} = v(s_{k−1}, s_{k}), e^{k} = e(s_{k−1}, s_{k}) for k = 1, 2, …, m. Because of the normalizations ⟨u^{k}, e^{k}⟩ = ⟨v^{k}, e^{k}⟩ = 1 we have

⟨u^{k}, v^{k}⟩ − 1 = ⟨u^{k} − e^{k}, v^{k} − e^{k}⟩

and hence

|B_{P}(s; u_{s}, v_{s})|^{2} = | ∑_{k=1}^{m} ⟨u^{k} − e^{k}, v^{k} − e^{k}⟩ |^{2} ≤ (∑_{k−1}^{m} ∥u^{k} − e^{k}∥ · ∥v^{k} − e^{k}∥)^{2}

≤ ∑_{k=1}^{m} ∥u^{k} − e^{k}∥^{2} ∑_{k=1}^{m} ∥v^{k} − e^{k}∥^{2}.

Now we can write

∥u^{k} − e^{k}∥^{2} = ∥u^{k}∥^{2} − 1 = ∥u(s_{k−1}, s_{k})∥^{2} − 1.

If k = 1 this is just ∥u_{t_{1}}∥^{2} − 1, and if k > 1 it becomes

\frac{∥u_{s_{k}}∥^{2}}{∥u_{s_{k−1}}∥^{2}} − 1 = ∥u_{s_{k−1}}∥^{−2}(∥u_{s_{k}}∥^{2} − ∥u_{s_{k−1}}∥^{2})
≤ ∥u_{s_{k}}∥^{2} − ∥u_{s_{k−1}}∥^{2}.

Thus we can estimate ∑ ∥u^{k} − e^{k}∥^{2} using a telescoping series to obtain

∑_{k=1}^{m} ∥u^{k} − e^{k}∥^{2} ≤ ∥u_{s_{m}}∥^{2} − 1 = ∥u_{s}∥^{2} − 1.

Similarly,

∑_{k=1}^{m} ∥v^{k} − e^{k}∥^{2} ≤ ∥v_{s_{m}}∥^{2} − 1 = ∥v_{s}∥^{2} − 1,

and the first of the two inequalities follows.

The proof of the second is similar, and we merely indicate the changes. It suffices to show that for any partition P of (0, t] which contains s, we have

|B_{P}(t; u_{t}, v_{t}) − B_{P∩(0,s]}(s; u_{s}, v_{s})| ≤ (∥u_{t}∥^{2} − ∥u_{s}∥^{2})(∥v_{t}∥^{2} − ∥v_{s}∥^{2}).

The desired inequality will follow by taking the limit on P. Suppose that

P = {0 = s_{0} < s_{1} < ⋯ < s_{m} = s = t_{0} < t_{2} < ⋯ < t_{n} = t}.

Then if we write out the formula for B_{P}(t; u_{t}, v_{t}) we find that

B_{P}(t; u_{t}, v_{t}) = B_{P∩(0,s]}(s; u_{s}, v_{s}) + ∑_{l=1}^{n}(⟨u(t_{l−1}, t_{l}), v(t_{l−1}, t_{l})⟩ − 1).

Thus we have to show that

| ∑_{l=1}^{n}(⟨u(t_{l−1}, t_{l}), v(t_{l−1}, t_{l})⟩ − 1)|^{2} ≤ (∥u_{t}∥^{2} − ∥u_{s}∥^{2})(∥v_{t}∥^{2} − ∥v_{s}∥^{2}).

But if we write u^{l} = u(t_{l−1}, t_{l}), v^{l} = v(t_{l−1}, t_{l}), e^{l} = e(t_{l−1}, t_{l}), then we have

⟨u^{l}, v^{l}⟩ − 1 = ⟨u^{l} − e^{l}, v^{l} − e^{l}⟩

and as in the proof of the first inequality it suffices to show that

∑_{l=1}^{n} ∥u^{l} − e^{l}∥^{2} ≤ ∥u_{t}∥^{2} − ∥u_{s}∥^{2},

with a similar estimate for u replaced with v. But noting that

{s = t_{0} < t_{1} < ⋯ < t_{n} = t}

is a partition of the interval (s, t], we can make similar estimates as those made in the preceding argument to estimate the sum

∑_{l=1}^{n} ∥u^{l} − e^{l}∥^{2} = ∑_{l=1}^{n}(∥u(t_{l−1}, t_{l})∥^{2} − 1)

with a telescoping series whose sum is ∥u_{t}∥^{2} − ∥u_{s}∥^{2} □

From the inequalities of Proposition 9.8 and Lemma 7.4, we immediately conclude that P^{e} : Δ^{2} → ℂ is continuous and vanishes at the origin.

Thus, to show that P^{e} is an e-logarithm (and therefore coincides with L^{e}) it remains only to show that for t > 0 and x, y ∈ D(t) we have

e^{P^{e}(t;x,y)} = \frac{⟨x, y⟩}{⟨x, e_{t}⟩ ⟨e_{t}, y⟩}.

Since both sides are homogeneous functions of degree zero in x and y, it suffices to prove the formula for x, y normalized so that ⟨x, e_{t}⟩ = ⟨y, e_{t}⟩ = 1. That is, we must prove that

(9.10)  e^{B_{∞}(t;x,y)} = ⟨x, y⟩ ,

for all x, y ∈ D(t) satisfying ⟨x, e_{t}⟩ = ⟨y, e_{t}⟩ = 1. We will deduce (9.10) from the following lemma, which may be considered a generalization of the familiar formula

lim_{n→∞}(1 + z/n)^{n} = e^{z}.

l^{1} (resp. l^{2}) will denote the space of all sequences of complex numbers z = (z(1), z(2), …) which are summable (resp. square summable). The norm of z ∈ l^{2} is denoted ∥z∥_{2}.

Lemma 9.11. Let I be a directed set and let {z_{α} : α ∈ I} be a net of sequences in l^{1} ∩ l^{2} satisfying

(9.11a)  lim_{α} ∥z_{α}∥_{2} = 0, and
(9.11b)  lim_{α} ∑_{k=1}^{∞} z_{α}(k) = ζ ∈ ℂ.

Then for every α ∈ I the infinite product ∏_{k=1}^{∞}(1 + z_{α}(k)) converges absolutely and we have

lim_{α} ∏_{k=1}^{∞}(1 + z_{α}(k)) = e^{ζ}.

Remarks. Notice that since ∑_{k=1}^{∞} |z_{α}(k)| converges for every α ∈ I, every infinite product p_{α} = ∏_{k=1}^{∞}(1 + z_{α}(k)) converges absolutely as well.

Note too that that Lemma 9.11 reduces to the familiar formula

lim_{n→∞}(1 + ζ/n)^{n} = e^{ζ},

by taking I = {1, 2, … } and

z_{n} = (\underbrace{ζ/n, ζ/n, …, ζ/n,}_{n times} 0, …).

In this case (9.11a) follows from the fact that ∥z_{n}∥_{2} = |ζ|/n^{1/2} → 0 as n → ∞, and (9.11b) is an identity for every n.

proof of Lemma 9.11. Let log denote the principal branch of the complex logarithm on the region {1 + z : |z| < 1}. Then for sufficiently large α we have

sup_{k≥1} |z_{α}(k)| ≤ ∥z_{α}∥_{2} < 1,

and for such an α log(1 + z_{α}(k)) is defined for every k = 1, 2, . . .. We will show that the series ∑_{k} log(1 + z_{α}(k)) is absolutely convergent for large α and, in fact,

(9.12)  lim_{α} ∑_{k=1}^{∞} log(1 + z_{α}(k)) = ζ.

The required conclusion follows after exponentiating (9.12).

Now since

lim_{z→0} \frac{log(1 + z) − z}{z^{2}} = 1/2 < 1,

we can find ϵ > 0 so that

| log(1 + z) − z| < |z|^{2}

for all z with |z| < ϵ. If α is large enough that ∥z_{α}∥_{2} < ϵ then we have

∑_{k=1}^{∞} | log(1 + z_{α}(k)) − z_{α}| ≤ ∑_{k=1}^{∞} |z_{α}(k)|^{2} < ϵ^{2}.

In particular, ∑_{k} | log(1 + z_{α}(k))| < ∞ because z_{α} ∈ l^{1}. Now we obtain (9.12) by noting that for large α,

| ∑_{k} log(1 + z_{α}(k)) − ζ| ≤ ∑_{k} | log(1 + z_{α}(k)) − z_{α}(k)| + | ∑_{k} z_{α}(k) − ζ|
≤ϵ^{2} + | ∑_{k} z_{α}(k) − ζ|,

and using the hypothesis (9.11b) to estimate the second term on the right. □

We apply Lemma 9.11 to the directed set I of all partitions P of the interval (0, t] as follows. For every

P = {0 = t_{0} < t_{1} < ⋯ < t_{n} = t}

we have from (9.8)

B_{P}(t; x, y) = ∑_{k=1}^{n} z_{P}(k)

where z_{P}(k) = ⟨x(t_{k−1}, t_{k}), y(t_{k−1}, t_{k})⟩ − 1. Now for 0 ≤ a < b ≤ t we have

| ⟨x(a, b), y(a, b)⟩ − 1| = | ⟨x(a, b) − e(a, b), y(a, b) − e(a, b)⟩ |
≤ ∥x(a, b) − e(a, b)∥ · ∥y(a, b) − e(a, b)∥.

Let ϕ, ψ : [0, t] → ℝ be the functions ϕ(0) = ψ(0) = 1, ϕ(s) = ∥x_{s}∥^{2}, ψ(s) = ∥y_{s}∥^{2} for 0 < s ≤ 1. Then ϕ and ψ are continuous monotone increasing, and an estimate like those in the proof of Theorem 9.8 shows that

∥x(a, b) − e(a, b)∥^{2} = ∥x(a, b)∥^{2} − 1 ≤ ϕ(b) − ϕ(a)

and

∥y(a, b) − e(a, b)∥^{2} = ∥y(a, b)∥^{2} − 1 ≤ ψ(b) − ψ(a).

Thus we can estimate |z_{P}(k)| as follows. Setting

ϵ(α) = sup_{|t−s|≤α} |ϕ(t) − ϕ(s)|

for positive α, we have

|z_{P}(k)|^{2} ≤ (ϕ(t_{k}) − ϕ(t_{k−1}))(ψ(t_{k}) − ψ(t_{k−1})) ≤ ϵ(|P|)(ψ(t_{k}) − ψ(t_{k−1})),

|P| denotinging the norm of P. It follows that

∑_{k=1}^{n} |z_{P}(k)|^{2} ≤ ϵ(|P|)(ϕ(t) − 1).

Since lim_{P} ϵ(|P|) = 0 because lim_{P} |P| = 0 and ϕ is uniformly continuous, we conclude that lim_{P} ∥z_{P}∥_{2} = 0. This establishes condition (9.11a). In this case (9.11b) is the formula

lim_{P} B_{P}(t; x, y) = B_{∞}(t; x, y).

Since 1 + z_{P}(k) = ⟨x(t_{k−1}, t_{k}), y(t_{k−1}, t_{k})⟩ we may conclude from Lemma 9.11 that

lim_{P} ∏_{I∈P} ⟨x_{I}, y_{I}⟩ = e^{B_{∞}(t;x,y)}.

But for every partition P we have

⟨x, y⟩ = ∏_{I∈P} ⟨x_{I}, y_{I}⟩ ,

and thus the preceding discussion implies that

⟨x, y⟩ = e^{B_{∞}(t;x,y)}

which is the required formula (9.10). □

Remark 9.13. The argument we have given yields a somewhat stronger result involving sequential convergence. For each t > 0, let P_{1,t} ≤ P_{2,t} ≤ … be any increasing sequence of partitions of the interval [0, t] with the property

lim_{n→∞} |P_{n,t}| = 0,

for every t. Then for every x, y ∈ D(t) satisfying ⟨x, e_{t}⟩ = ⟨y, e_{t}⟩ = 1, we claim that

(9.14)  L^{e}(t; x, y) = lim_{n→∞} ∑_{I∈P_{n,t}}(⟨x_{I}, y_{I}⟩ − 1).

To see that, fix x, y and define B′_{n}(t; x, y) by

B′_{n}(t; x, y) = ∑_{I∈P_{n,t}}(⟨x_{I}, y_{I}⟩ − 1).

(9.5) and (9.7) imply that the limit

B′_{∞}(t; x, y) = lim_{n→∞} B′_{n}(t; x, y)

exists. Moreover, we may apply Lemma 9.11 as in the preceding argument to conclude that

⟨x, y⟩ = e^{B′_{∞}(t;x,y)}.

Finally, the estimates of Proposition 9.8 are valid for B′_{∞}(t; x, y) as well as for B_{∞}(t; x, y). Thus for any pair of left-decomposable sections u, v satisfying ⟨u_{s}, e_{s}⟩ = ⟨v_{s}, e_{s}⟩ = 1, the function

t ∈ (0, ∞) ↦ B′_{∞}(t; u_{t}, v_{t})

is continuous and tends to 0 as t → 0+. It follows that

B′_{∞}(t; u_{t}, v_{t}) = B_{∞}(t; u_{t}, v_{t}) = L^{e}(t; u_{t}, v_{t})

for all t. Fixing t and choosing u, v appropriately, we obtain (9.14).

We will make use of formula (9.14) in the sequel.

10. Existence of measurable propagators.

The reader may have noticed that none of the results of sections 5 through 9 made any reference to measurability or to measurable sections, even though there is a natural Borel structure on the product system E. Measurability was simply not an issue in those matters. On the other hand, in the applications that will be discussed in part III it will be necessary do deal with measurable elements of D, and with a measurable reference section e; this is necessary in order to satisfy the measurability hypothesis of Definition 2.2. The existence of sufficiently many measurable elements of D is established in the following result.

Theorem 10.1. Let t_{0} > 0 and let u ∈ E(t_{0}) be a nonzero decomposable vector. Then there is a left-coherent decomposable Borel section t ∈ (0, ∞) → x_{t} ∈ D(t) such that x_{t_{0}} is a scalar multiple of u.

proof. We may assume without loss of generality that ∥u∥ = 1. Since u is decomposable, we can find elements a_{s}, u_{s} ∈ E(s) such that

(10.2)  u = a_{s}u_{t_{0}−s},  0 < s < t_{0};

and by renormalizing again if necessary we can arrange that ∥a_{s}∥ = ∥u_{s}∥ = 1. Set a_{t_{0}} = u. Then {a_{s} : 0 < s ≤ t_{0}} is a left-decomposable family of unit vectors. The following result implies that we can choose a in a measurable way.

Proposition 10.3. Let t ∈ (0, t_{0}] ↦ a_{t} ∈ E(t) be any left-coherent family of unit vectors. Then there is a function t ∈ (0, t_{0}] ↦ λ_{t} ∈ ℂ such that |λ_{t}| = 1 for every t and t ↦ λ_{t}a_{t} is a measurable section of E ↾_{(0,t_{0}]}.

Let us assume, for the moment, that the technical result 10.3 has been established. If we replace a_{s} with λ_{s}a_{s} in (10.2) then we may assume that a_{s} is measurable in s. We can now define a measurable section t ∈ (0, ∞) ↦ x_{t} ∈ E(t) by

x_{t} = \{ \begin{matrix}⟨a_{t}, e_{t}⟩^{−1} a_{t}, & for 0 < t ≤ t_{0} \\ a_{t_{0}}e(t_{0}, t), & for t > t_{0}, \end{matrix}

where {e(s, t) : 0 < s < t < ∞} is the propagator associated with e. It is clear that x ∈ D^{e} and that x_{t_{0}} is a scalar multiple of u, completing the proof of Theorem 10.1.

proof of 10.3. Consider the family of operators {P_{s} : 0 < s < t_{0}} ⊆ B(E(t_{0})) defined by

P_{s}(ξ) = a_{s} · L_{a_{s}}^{∗}(ξ), ξ ∈ E(t_{0})

(see the proof of Theorem 2.1 for a discussion of operators of the form L_{z}^{∗}, z ∈ E). Notice that since E(t_{0}) is spanned by E(s)E(t_{0} − s), P_{s} is uniquely determined by its action on decomposable vectors ξ_{1}ξ_{2}, ξ_{1} ∈ E(s), ξ_{2} ∈ E(t_{0} − s):

P_{s}(ξ_{1}ξ_{2}) = ⟨ξ_{1}, a_{s}⟩ a_{s}ξ_{2}.

This also shows that if we identify E(t_{0}) with E(s) ⊗ E(t_{0} − s) then P_{s} becomes

[a_{s}] ⊗ 1_{t_{0}−s},

[a_{s}] denoting the projection onto the one-dimensional subspace ℂ · a_{s} of E(s) and 1_{r} denoting the identity operator in B(E(r)). Now since {a_{s} : 0 < s < t_{0}} is left-coherent, the family of projections {P_{s} : 0 < s < t_{0}} satisfies

s_{1} < s_{2} ⟹ P_{s_{2}} ≤ P_{s_{1}}

(see the proof of Theorem 2.1). Thus for any fixed ξ ∈ E(t_{0}),

s ↦ ⟨P_{s}ξ, ξ⟩

is a monotone decreasing function of (0, t_{0}), and hence measurable. By polarization it follows that ⟨P_{s}ξ, η⟩ is measurable in s for any ξ, η ∈ E(t_{0}). Hence

s ∈ (0, t_{0}) ↦ P_{s} ∈ B(E(t_{0}))

is a measurable projection-valued operator function.

For each 0 < s < t_{0}, let Q_{s} denote the rank-on projection Q_{s} = [a_{s}] ∈ B(E(s)). We have just seen that

(10.5)  s ↦ Q_{s} ⊗ 1_{t_{0}−s}

is measurable and we claim now that Q itself is measurable. Equivalently, we claim that for any pair of measurable sections ξ_{s}, η_{s} ∈ E(s), the complex valued function s ∈ (0, t_{0}) ↦ ⟨Q_{s}ξ_{s}, η_{s}⟩ is measurable. To see that we choose any measurable section t ↦ u_{t} ∈ E(t) of unit vectors and write

⟨Q_{s}ξ_{s}, η_{s}⟩ = ⟨Q_{s}ξ_{s}, η_{s}⟩ ⟨u_{t_{0}−s}, u_{t_{0}−s}⟩ = ⟨Q_{s} ⊗ 1_{t_{0}−s}(ξ_{s}u_{t_{0}−s}), η_{s}u_{t_{0}−s}⟩ ,

for 0 < s < t_{0}. The right side is measurable in s by (10.6) and the fact that both ξ_{s}u_{t_{0}−s} and η_{s}u_{t_{0}−s} are measurable functions of s.

Finally, since s ∈ (0, t_{0}) ↦ Q_{s} ∈ B(E(s)) is a measurable family we claim that there is a measurable section s ∈ (0, t_{0}) ↦ b_{s} ∈ E(s) satisfying ∥b_{s}∥ = 1 for every s and

(10.6)  Q_{s}b_{s} = b_{s},  0 < s < t_{0}.

To prove (10.6) we choose a measurable basis for the family of Hilbert spaces p : E → (0, ∞). That is, we find a sequence of measurable sections e_{n} : t ∈ (0, ∞) ↦ e_{n}(t) ∈ E(t), n = 1, 2, … such that {e_{1}(t), e_{2}(t), … } is an orthonormal basis for E(t) for every t > 0. This is possible because of the last axiom for product systems [2, (1.8) iii]. For every t > 0 we must have Q_{t}e_{n}(t) ≠ 0 for some n, and we define n(t) to be the smallest such n. Now for every positive integer k we have

{t ∈ (0, ∞) : n(t) > k} = ⋂_{i=1}^{k}{t ∈ (0, ∞) : ⟨Q_{t}e_{i}(t), e_{i}(t)⟩ = 0},

and the right side is a Borel set since each function t ↦ ⟨Q_{t}e_{i}(t), e_{i}(t)⟩ is measurable. It follows that the function t ∈ (0, ∞) ↦ n(t) ∈ ℝ is measurable. Hence ξ_{t} = e_{n(t)} defines a measurable section of E having the property that Q_{t}ξ_{t} ≠ 0 for every t > 0. Thus we obtain a section b as required by (10.6) by setting

b_{t} = ∥Q_{t}ξ_{t}∥^{−1}Q_{t}ξ_{t}, 0 < t < t_{0}.

Now since Q_{t} is the projection onto the one-dimensional space ℂa_{t} it follows that there is a complex number λ_{t} such that b_{t} = λ_{t}a_{t}. Since ∥a_{t}∥ = ∥b_{t}∥ = 1 we have |λ_{t}| = 1, completing the proof of Proposition 10.3. □

Part III. Applications

In the following two sections we apply the preceding results to classify certain product systems and certain E_{0}-semigroups.

11. Decomposable continuous tensor products. A product system p : E → (0, ∞) is said to be decomposable if for every t > 0, E(t) is the closed linear span of the set D(t) of all its decomposable vectors. It is easy to see that if this condition is satisfied for a single t_{0} > 0 then it is satisfied for every t > 0.

Theorem 11.1. A decomposable product system is either isomorphic to the trivial product system with one-dimensional spaces E(t), t > 0, or it is isomorphic to one of the standard product systems E_{n}, n = 1, 2, …, ∞.

Remarks. We recall that any product system with one-dimensional fibers E(t) for every t > 0 is isomorphic to the trivial product system p : Z → (0, ∞), where Z = (0, ∞) × ℂ with multiplication (s, z)(t, w) = (s + t, zw), with the usual inner product on ℂ, and with projection p(t, z) = t [6, Corollary of Prop. 2.3] and [7].

proof of Theorem 11.1.

By Theorem 10.3, we can find a Borel section t ∈ (0, ∞) ↦ e(t) ∈ D(t). By replacing e_{t} with e_{t}/∥e_{t}∥ if necessary we may assume that ∥e_{t}∥ = 1 for every t. Let L^{e} : Δ^{2} → ℂ be the function provided by Theorem 8.2 which satisfies

(11.2)  e^{L^{e}(x,y)} = \frac{⟨x, y⟩}{⟨x, e_{t}⟩ ⟨e_{t}, y⟩}

for every x, y ∈ D(t), t > 0. We will show first that (Δ, L^{e}) is a metric path space; that is, L^{e} is an additive form on Δ^{2}.

Indeed, most of that assertion follows immediately from the results of Parts I and II. Theorem 9.2 implies that L^{e} restricts to a positive definite function on Δ(t) × Δ(t) for every t > 0, and because of Propositions 9.5 and 9.7 taken together with with formula 9.14, L^{e} must satisfy the separability condition of Definition 2.6. Proposition 8.8 shows that L^{e} is additive with defect function ψ : Δ × Δ → ℂ of the form ψ(x, y) = ψ_{s}(t; y) for every x ∈ D(s), y ∈ D(t), s, t > 0. Thus we need only establish the measurability criterion of Definition 2.2.

Notice that Definition 2.2 makes reference to the propagator {ż(r, s) : 0 ≤ r < s ≤ t} associated with an element ż ∈ Δ(t). While this propagator in the path space Δ is uniquely determined by ż, an element z ∈ D(t) does not determine a unique propagator {z(r, s) ∈ D(s − r) : 0 ≤ r < s ≤ t} because of the failure of unique factorization in {D(t) : t > 0}. Nevertheless, any propagator in Δ can be lifted to a measurable propagator in {D(t) : t > 0}. More precisely, given any element z ∈ D(t) then Theorem 10.1 provides a left-coherent measurable family {z_{s} ∈ D(s) : 0 < s ≤ t} for which z_{t} is a scalar multiple of z. Recalling that a leftcoherent family gives rise to a unique propagator because of the left cancellation law in {D(t) : t > 0}, we conclude that the propagator {z(r, s) ∈ D(s − r) : 0 ≤ r < s ≤ t} associated with {z_{s} : 0 < s ≤ t} projects to the required propagator in Δ, viz

ż_{s} = ż_{r}ż(r, s), 0 < r < s ≤ t,

and of course ż(0, s) = ż_{s} for 0 < s ≤ t.

In order to establish the measurability criterion of Definition 2.2, choose T_{1}, T_{2} satisfying 0 < T_{1} < T_{2} and choose x ∈ D(T_{1}), y ∈ D(T_{2}). By the preceding remarks, we may find a measurable left-coherent section y_{s} ∈ D(s), 0 < s ≤ T_{2} whose propagator projects to the propagator of ẏ ∈ Δ(T_{2}). It suffices to show that the function

(11.3)  λ ∈ (0, T_{2} − T_{1}) ↦ L^{e}(T_{1}; x, y(λ, λ + T_{1}))

is a complex-valued Borel function.

To prove (11.3) we will make use of the fact that L^{e} is the limit of a sequence of functions for which the fact of measurability is obvious. Let P_{1}, P_{2}, … be a sequence of finite partitions of the interval [0, T_{1}] such that P_{n+1} is a refinement of P_{n} and such that the norms |P_{n}| tend to zero as n → ∞. By (9.14) we may conclude that for every u, v ∈ D(T_{1}) satisfying ⟨u, e_{T_{1}}⟩ = ⟨v, e_{T_{1}}⟩ = 1, we have

L^{e}(T_{1}; u, v) = lim_{n→∞} ∑_{I∈P_{n}}(⟨u_{I}, v_{I}⟩ − 1).

Taking

u = ⟨x, e_{T_{1}}⟩^{−1} x  and
v =v_{λ} = ⟨y(λ, λ + T_{1}), e_{T_{1}}⟩^{−1} y(λ, λ + T_{1}),

we see that the left side of (11.3) is exhibited as the limit of a convergent sequence of functions

f_{n}(λ) = ∑_{I∈P_{n}}(⟨u_{I}, (v_{λ})_{I}⟩ − 1).

Thus it suffices to show that each f_{n} is Borel-measurable on 0 < λ < T_{2} − T_{1}. In order to see that, choose an interval I = (a, b] ⊆ (0, T_{1}] for which a < b and look at the inner product ⟨u_{I}, (v_{λ})_{I}⟩. Noting that

y(λ, λ + T_{1}) =y(λ, λ + a)y(λ + a, λ + b)y(λ + b, λ + T_{1})
e_{T_{1}} =e_{a}e(a, b)e(b, T_{1}),

we can write down an obvious propagator for v_{λ} and we find that

v_{λ}(a, b) = ⟨y(λ + a, λ + b), e(a, b)⟩^{−1} y(λ + a, λ + b).

It follows that f_{n}(λ) is a finite linear combination of functions of the form

(11.4)  λ ∈ (0, T_{2} − T_{1}) ↦ \frac{⟨u(a, b), y(λ + a, λ + b)⟩}{⟨e(a, b), y(λ + a, λ + b))⟩} − 1.

Now for any element w ∈ D(b − a) we have

⟨y_{λ+a}w, y_{λ+b}⟩ = ⟨y_{λ+a}w, y_{λ+a}y(λ + a, λ + b)⟩
= ∥y_{λ+a}∥^{2} ⟨w, y(λ + a, λ + b)⟩ .

Thus the right side of (11.4) can be written

\frac{⟨y_{λ+a}u(a, b), y_{λ+b}⟩}{⟨y_{λ+a}e(a, b), y_{λ+b}⟩} − 1.

This is obviously a measurable function of λ because s ↦ y_{s} is a measurable section, left multiplication by a fixed element of E is a measurable mapping of E into itself, and because the inner product ⟨·, ·⟩ : E^{2} → ℂ is measurable

Thus we have established the fact that (Δ, L^{e}) is a metric path space. By Theorem 4.3 there is a separable Hilbert space C, a function ρ : Δ → ℂ, and a logarithm mapping

log : Δ → P_{C}

such that log(xy) = log(x) ⊞ log(y) for every x, y ∈ Δ and

(11.5)  L^{e}(t; x_{1}, x_{2}) = ⟨log(x_{1}), log(x_{2})⟩ + ρ(x_{1}) + \overline{ρ(x_{2})}

for all x_{1}, x_{2} ∈ Δ(t), t > 0. In fact, in order to obtain the necessary measurability properties, we must use the specific function ρ defined in the proof of Theorem 4.3.

Now when the path space P_{C} is exponentiated, it gives rise to the standard product system E_{C}. In more detail, consider the symmetric Fock space H_{C} over the one-particle space L^{2}((0, ∞); C), and consider the exponential map

exp : L^{2}((0, ∞); C) → H_{C}

defined by

exp(f) = ∑_{n=0}^{∞} \frac{1}{\sqrt{n!}}f^{⊗n}.

For every t > 0 we define

E_{C}(t) = \overline{span}\{exp(f) : f ∈ P_{C}(t)\}.

E_{C} is the total space of this family of Hilbert spaces, with multiplication ξ ∈ E_{C}(s), η ∈ E_{C}(t) ↦ ξη ∈ E_{C}(s+t) defined uniquely by requiring that the generating vectors should multiply thus:

exp(f) exp(g) = exp(f ⊞ g), f ∈ P_{C}(s), g ∈ P_{C}(t).

We will use (11.5) to construct an isomorphism of product systems W : E → E_{C}. But in order to define W it is necessary to solve another cohomological problem. The result is summarized as follows. We write

p : D = {(t, x) : x ∈ D(t), t > 0} → (0, ∞)

for the fiber space determined by the family of sets D(t), t > 0 with projection p(t, x) = t. Notice that D carries a natural Borel structure as a subspace of E.

Theorem 11.6. There is a Borel-measurable function u : (0, ∞) → ℂ satisfying |u(t)| = 1 for every t > 0, such that the function f : D → ℂ defined by

f(x) = u(t) ⟨x, e_{t}⟩ e^{ρ(x)}, x ∈ D(t), t > 0

is multiplicative: f(xy) = f(x)f(y), x ∈ D(s), y ∈ D(t).

proof of Theorem 11.6. Let f_{0}(x) = ⟨x, e_{t}⟩ e^{ρ(x)}. Notice first that f_{0} is Borelmeasurable. Indeed, recalling the formula for ρ(x) given in the proof of Theorem 4.3, we have for every x ∈ D(s) and s > 0,

(11.7)  ρ(x) = ⟨[x] − [e_{s}], ϕ_{s}⟩ + L^{e}(s; x, e_{s}) − \frac{1}{2}(L^{e}(s; e_{s}, e_{s}) + ∥ϕ_{s}∥^{2}).

Because of the representation of L^{e} as a sequential limit in (9.14) we see that L^{e} : D^{2} → ℂ is a Borel function, and it follows that x ∈ D(s) ↦ [x] − [e_{s}] defines a Borel map of D into the Hilbert space H_{∞}. Finally, since s ∈ (0, ∞) ↦ ϕ_{s} is a Borel function taking values in the space of all locally square integrable C-valued functions on (0, ∞), we see that the right side of (11.7) defines a Borel function on the subspace D of E. Since the inner product is a Borel function on E, it follows that

f_{0}(x) = ⟨x, e_{s}⟩ e^{ρ(x)}, x ∈ D(s)

defines a complex-valued Borel function on D.

We consider the associated “coboundary” c : D × D → ℂ:

c(x, y) = \frac{f_{0}(xy)}{f_{0}(x)f_{0}(y)}.

Note that c is a Borel function as well, since the multiplication operation of E is Borel measurable. We will show that for every s, t > 0, x_{i} ∈ D(s), y_{i} ∈ D(t),

(11.8.1)  |c(x_{1}, y_{1})| =1,
(11.8.2)  c(x_{1}, y_{1}) =c(x_{2}, y_{2}).

Assume for the moment that the equations (11.8) have been established. It follows that there is a function

c_{0} : (0, ∞) × (0, ∞) → {z ∈ ℂ : |z| = 1}

such that for all x ∈ D(s), y ∈ D(t) we have

c(x, y) = c_{0}(s, t).

c_{0} is clearly measurable because c_{0}(s, t) = c(e_{s}, e_{t}), and r ↦ e_{r} ∈ D(r) is a measurable section. We will then show that c_{0} satisfies the multiplier equation,

(11.9)  c_{0}(r, s + t)c_{0}(s, t) = c_{0}(r + s, t)c_{0}(r, s),

for every r, s, t > 0. By [6, Corollary of Propostion 2.3], there is a Borel function u : (0, ∞) → {z ∈ ℂ : |z| = 1} such that

c_{0}(s, t) = \frac{u(s)u(t)}{u(s + t)}, s, t > 0.

Once we have u it is clear that the function f(x) = u(t)f_{0}(x), x ∈ D(t), t > 0 satisfies

f(xy) = f(x)f(y).

Thus, we must prove the formulas (11.8) and (11.9). For t > 0, consider the function W_{t} : D(t) → E_{C}(t) defined by

W_{t}(x) = f_{0}(x) exp log(x) = ⟨x, e_{t}⟩ e^{ρ(x)} exp log(x).

We claim that for x_{1}, x_{2} ∈ D(t) we have

(11.10)  ⟨W_{t}(x_{1}), W_{t}(x_{2})⟩ = ⟨x_{1}, x_{2}⟩ .

Indeed, using Theorem 4.3 we have

⟨exp log(x_{1}), exp log(x_{2})⟩ =e^{⟨log(x_{1}),log(x_{2})⟩} = exp(L^{e}(t; x_{1}, x_{2}) − ρ(x_{1}) − \overline{ρ(x_{2})})
=\frac{⟨x_{1}, x_{2}⟩}{⟨x_{1}, e_{t}⟩ ⟨e_{t}, x_{2}⟩}e^{−(ρ(x_{1})+\overline{ρ(x_{2})})}.

Thus the left side of (11.10) is

f_{0}(x_{1})\overline{f_{0}(x_{2})} ⟨exp log(x_{1}), exp log(x_{2})⟩ = ⟨x_{1}, e_{t}⟩ \overline{⟨x_{2}, e_{t}⟩}\frac{⟨x_{1}, x_{2}⟩}{⟨x_{1}, e_{t}⟩ ⟨e_{t}, x_{2}⟩} = ⟨x_{1}, x_{2}⟩ ,

as asserted.

The family {W_{t} : t > 0} obeys the following multiplicative rule. For x ∈ D(s), y ∈ D(t), we claim

(11.11)  W_{s+t}(xy) = c(x, y)W_{s}(x)W_{t}(y).

Notice that the multiplication on the left side of (11.11) is performed in E and on the right side it is performed in E_{C}. To check (11.11), recall that the multiplication in E_{C} is related to the operation ⊞ by the following formula: if f ∈ E_{C}(s) and g ∈ E_{C}(t) then

exp(x) exp(g) = exp(f ⊞ g).

Thus, using the additivity property of the logarithm mapping the left side of (11.11) can be rewritten

f_{0}(xy) exp log(xy) =f_{0}(xy) exp(log(x) ⊞ log(y)) = f_{0}(xy) exp(log(x)) exp(log(y))
=\frac{f_{0}(xy)}{f_{0}(x)f_{0}(y)}W_{s}(x)W_{t}(y) = c(x, y)W_{s}(x)W_{t}(y).

We claim next that for x_{i} ∈ D(s), y_{i} ∈ D(t) we have

(11.12)  c(x_{1}, y_{1})\overline{c(x_{2}, y_{2})} = 1.

To see this, choose x_{i} ∈ D(s), y_{i} ∈ D(t). Noting that by (11.10)

⟨x_{1}y_{1}, x_{2}y_{2}⟩ = ⟨x_{1}, x_{2}⟩ ⟨y_{1}, y_{2}⟩ = ⟨W_{s}(x_{1}), W_{s}(x_{2})⟩ ⟨W_{t}(y_{1}), W_{t}(y_{2})⟩
= ⟨W_{s}(x_{1})W_{t}(y_{1}), W_{s}(x_{2})W_{t}(y_{2})⟩ ,

we have

c(x_{1}, y_{1})\overline{c(x_{2}, y_{2})} ⟨x_{1}y_{1}, x_{2}y_{2}⟩ =
c(x_{1}, y_{1})\overline{c(x_{2}, y_{2})} ⟨W_{s}(x_{1})W_{t}(y_{1}), W_{s}(x_{2})W_{t}(y_{2})⟩ =
⟨c(x_{1}, y_{1})W_{s}(x_{1})W_{t}(y_{1}), c(x_{2}, y_{2})W_{s}(x_{2})W_{t}(y_{2})⟩ =
⟨W_{s+t}(x_{1}y_{1}), W_{s+t}(x_{2}y_{2})⟩ = ⟨x_{1}y_{1}, x_{2}y_{2}⟩ .

The claim follows after cancelling ⟨x_{1}y_{1}, x_{2}y_{2}⟩ ≠ 0.

Set x_{2} = x_{1}, y_{2} = y_{1} in (11.12) to obtain |c(x_{1}, y_{1})| = 1. Thus if we multiply through in (11.12) by c(x_{2}, y_{2}) we obtain

c(x_{1}, y_{1}) = c(x_{2}, y_{2}),

hence (11.8.1) and (11.8.2) are established.

Now we can define a function c_{0} : (0, ∞) × (0, ∞) → ℂ by c_{0}(s, t) = c(e_{s}, e_{t}). c_{0} is a Borel function which, because of (11.11), obeys

W_{s+t}(xy) = c_{0}(s, t)W_{s}(x)W_{t}(y), x ∈ D(s), y ∈ D(t).

The latter formula implies that c_{0} must satisfy the multiplier equation (11.9). Indeed, for r, s, t > 0 and x ∈ D(r), y ∈ D(s), z ∈ D(t) we have

W_{r+s+t}(x(yz)) = c_{0}(r, s + t)W_{r}(x)W_{s+t}(yz) = c_{0}(r, s + t)c_{0}(s, t)W_{r}(x)W_{s}(y)W_{t}(z),

while

W_{r+s+t}((xy)z) = c_{0}(r+s, t)W_{r+s}(xy)W_{t}(z) = c_{0}(r+s, t)c_{0}(r, s)W_{r}(x)W_{s}(y)W_{t}(z).

Since W_{r}(x)W_{s}(y)W_{t}(z) ≠ 0, equation (11.9) follows.

The argument given above can now be applied to complete the proof of Theorem 11.6 □

We are now in position to write down an isomorphism E ≅ E_{C}. Choose u as in Theorem 11.6. For every t > 0, x ∈ D(t) define W_{t}(x) ∈ E_{C}(t) by

W_{t}(x) = u(t) ⟨x, e_{t}⟩ e^{ρ(x)} exp(log(x)).

Equation (11.10) implies that

⟨W_{t}(x_{1}), W_{t}(x_{2})⟩ = ⟨x_{1}, x_{2}⟩ .

Since E(t) is spanned by D(t), W_{t} can be extended uniquely to a linear isometry of E(t) into E_{C}(t), and we will denote the extended mapping by the same letter W_{t}. The range of W_{t} is given by

W_{t}(E(t)) = \overline{span}[exp log(D(t))] = E_{C}(t),

because the set log(D(t)) is a strongly spanning subset of P_{C}(t). Thus the total map

W : E → E_{C}

is an isomorphism of families of Hilbert spaces.

Because of the way we chose the function u, the multiplication formula (11.11) simplifies to

W_{s+t}(xy) = W_{s}(x)W_{t}(y), x ∈ D(s), y ∈ D(t).

Using bilinearity and the fact that D(r) spans E(r) for every r > 0, the latter implies that W is a homomorphism of product structures in that

W_{s+t}(ξη) = W_{s}(ξ)W_{t}(η), ξ ∈ E(s), y ∈ E(t).

In particular, W is a bijection of the standard Borel space E onto the standard Borel space E_{C}. Thus to see that W is a Borel isomorphism (and therefore an isomorphism of product systems), it suffices to show that it is measurable. The proof of that is a routine variation on the argument presented in detail in [2, pp 55–57], and we omit it.

Finally, notice that it is possible that C is the trivial Hilbert space {0}. However, in this case E_{C} is the trivial product system with one-dimensional fibers. If C ≠ {0} and we let n be the dimension of C, then n = 1, 2, …, ℵ_{0} and E_{C} is the standard product system E_{n} □

12. Decomposable E_{0}-semigroups. Let α = {α_{t}; t ≥ 0} be an E_{0}-semigroup, and for every t > 0 let E(t) be the operator space

E(t) = {T ∈ B(H) : α_{t}(A)T = T A, ∀A ∈ B(H)}.

E(t) is a Hilbert space relative to the inner product defined on it by

T^{∗}S = ⟨S, T ⟩ 1.

The family of Hilbert spaces

p : E = {(t, T) : t > 0, T ∈ E(t)} → (0, ∞)

with projection p(t, T) = t is actually a product system with respect to operator multiplication (s, S)(t, T) = (s+t, ST) [2]. In particular, E(s+t) is the norm closed linear span of the set of all products {ST : S ∈ E(s), T ∈ E(t)}.

As with any product system, it makes sense to speak of decomposable elements of E(t); thus, an operator T ∈ E(t) is decomposable if, for every 0 < s < t, T admits a factorization T = AB where A ∈ E(s) and B ∈ E(t − s). The set of all decomposable operators in E(t) is denoted D(t). Let H(t) = [D(t)H] be the closed linear span of the ranges of all operators in D(t). The spaces H(t) are obviously decreasing with t.

Definition 12.1. α = {α_{t} : t ≥ 0} is called decomposable if [D(t)H] = H for some (and therefore every) t > 0.

The following simple result shows that this terminology is consistent with the notion of decomposability for product systems.

Proposition 12.2. An E_{0}-semigroup α is decomposable iff its associated product system is decomposable in the sense of section 11.

proof. Suppose first that E is a decomposable product system. Then for every t > 0 E(t) is the norm-closed linear span of D(t). Since we have [E(t)H] = H for the product system of an arbitrary E_{0}-semigroup, it follows that H = [D(t)H] as well.

Conversely, assuming that α satisfies Definition 12.1 we pick t > 0 and an operator T ∈ E(t) such that T is orthogonal to D(t). Because of the definition of the inner product in E(t) it follows that for every S ∈ D(t) we have

T^{∗}S = ⟨S, T ⟩ 1 = 0,

hence T H is orthogonal to [D(t)H] = H, hence T = 0. Thus D(t) spans E(t) □

We can immediately deduce from Theorem 11.1 that the product system of a decomposable E_{0}-semigroup is either the trivial product system or it is isomorphic to one of the standard product systems E_{n}, n = 1, 2, …, ∞. Since the product system of an E_{0}-semigroup is a complete invariant for cocycle conjugacy and since the standard product systems E_{n} are associated with CCR flows (or CAR flows) [2], we can infer the following classification of E_{0}-semigroups as an immediate consequense of Theorem 11.1.

Theorem 12.3. Let α be a decomposable E_{0}-semigroup acting on B(H) which is nontrivial in the sense that it cannot be extended to a group of automorphisms of B(H). Then α is cocycle conjugate to a CCR flow.

Concluding remarks. Note that Theorem 12.3 implies that every decomposable E_{0}- semigroup α has plenty of intertwining semigroups...that is, semigroups of isometries U = {U_{t} : t ≥ 0} acting on H for which

α_{t}(A)U_{t} = U_{t}A, A ∈ B(H), t > 0.

Indeed, a decomposable E_{0}-semigroup must be completely spatial.

In a more philosophical vein, we conclude that any construction of E_{0}-semigroups which starts with a path space cannot produce anything other than CCR flows and their cocycle perturbations. For example, in [2 pp 14–16] we gave examples of product systems using Gaussian random processes and Poisson processes. The latter examples did not appear to contain enough units to be standard ones. However, a closer analysis showed that there were “hidden” units, and in fact there were enough of them so that these product systems were indeed standard. Theorems 11.1 and 12.4 serve to clarify this phenomenon because all examples constructed in this way from random processes are obviously decomposable.

We believe that Theorem 12.4 is analogous to the familiar description of representations of the compact operators (i.e., every representation is unitarily equivalent to a multiple of the identity representation), or to the Stone-von Neumann theorem. Certainly, the conclusion of 12.4 implies that decomposable E_{0}-semigroups exhibit “type I” behavior. What is interesting here is that the E_{0}-semigroups whose product systems are isomorphic to a given one E correspond bijectively with the “essential” representations of the spectral C^{∗}-algebra C^{∗}(E) [3, section 3], [4], and [5]. This correspondence has the feature that unitary equivalence of representations of C^{∗}(E) corresponds to conjugacy of E_{0}-semigroups. More precisely, if π_{1} and π_{2} are two essential representations of C^{∗}(E) with corresponding E_{0}-semigroups α_{1} and α_{2}, then α_{1} and α_{2} are conjugate iff there is a gauge automorphism γ of C^{∗}(E) such that π_{2} is unitarily equivalent to π_{1} ∘ γ. The structure of the group of gauge automorphisms has been explicitly calculated for the standard examples C^{∗}(E_{n}), n = 1, 2, …, ∞ [2, section 8].

In the case of decomposable E_{0}-semigroups the spectral C^{∗}-algebra is a standard one C^{∗}(E_{n}), n = 1, 2, …, ∞. These examples are continuous analogues of the Cuntz algebra O_{∞} [9], and are far from being type I C^{∗}-algebras. Nevertheless, if we agree to identify two representations of C^{∗}(E_{n}) up to unitary equivalence modulo cocycle perturbations (that is to say, up to cocycle perturbations of the associated E_{0}-semigroups), as well as up to the internal action of the gauge group, then the resulting set of equivalence classes of representations is smooth: it is parameterized by a single integer n = 1, 2, …, ∞. The integer n is, of course, the numerical index of the associated E_{0}-semigroup.

On the other hand, we remind the reader that Powers has constructed examples of E_{0}-semigroups that are of type II (they have some intertwining semigroups but not enough of them) [27], and others that are of type III (they have no intertwining semigroups whatsoever) [25]. None of these more exotic E_{0}-semigroups can be decomposable. The structure of the product systems associated with such E_{0}- semigroups remains quite mysterious.

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