E_{0}-SEMIGROUPS IN QUANTUM FIELD THEORY

William Arveson

Department of Mathematics University of California Berkeley CA 94720, USA

8 July 1994

Abstract. A lecture presented at the von Neumann symposium on Quantization and Nonlinear Wave Equations held at MIT in June, 1994. We describe the role of semigroups of endomorphisms of von Neumann algebras in algebraic formulations of quantum field theory, and present a summary of recent developments in the theory of E_{0}-semigroups.

1991 Mathematics Subject Classification. Primary 46L40; Secondary 81E05. Key words and phrases. von Neumann algebras, semigroups, automorphism groups, quantum

field theory. This research was supported in part by NSF grant DMS92-43893

Typeset by AMS-TEX

1. Relation to symmetry groups of algebras of local observables. In the algebraic formulation of quantum field theory, one works with a C^{∗}-algebra

A that is acted upon by a group of automorphisms representing the symmetries of spacetime (for example the Poincaré group), and which is also given a local structure (subalgebras of local observables). This local structure normally consists of associating a (von Neumann) subalgebra of A to every bounded region of spacetime in a coherent way, so that inclusions match up correctly, so that these subalgebras transform covariantly under the group action, so that the topologies fit together consistently, and which may satisfy additional conditions arising from the underlying physics. This C^{∗}-algebra is not necessarily presented in a representation on a Hilbert space. However, given any state of A which restricts to a normal state on the local subalgebras, one can perform the usual GNS construction and obtain a representation of this algebra on a Hilbert space H.

Typically, the group of automorphisms (or large subgroups of it) leave the state invariant, and therefore one can naturally construct a unitary representation of the group on H which implements the action. Moreover, once one has a concrete representation of A, one can associate von Neumann algebras with every (not necessarily bounded) open subset of spacetime by taking weak closures of appropriate unions.

For example, there is a von Neumann algebra M(C) ⊆ B(H) associated with the forward light cone

C = {(t, x) ∈ ℝ × ℝ^{3} : (x_{1}^{2} + x_{2}^{2} + x_{3}^{2})^{1/2} < t }.

If v is any vector of ℝ × ℝ^{3} which lies inside C then the one parameter group λ ∈ ℝ ↦ λv defines a physical flow of time. This flow is represented by a oneparameter unitary group {U_{λ} : λ ∈ ℝ } acting on H. Since C + λv ⊆ C for λ ≥ 0, we may conclude from the covariance properties of the given structure that for nonnegative λ we have

(1.1)  U_{λ}M(C)U_{λ}^{∗} ⊆ M(C).

Thus (1.1) defines a semigroup of isometric unit-preserving ∗-endomorphisms which acts on the von Neumann algebra M(C). In the simplest case where M(C) is a factor of type I_{∞}, we have a prototypical example of an E_{0}-semigroup {α_{λ} : λ ≥ 0 }

α_{λ}(A) = U_{λ}AU_{λ}^{∗}, A ∈ M(C), λ ≥ 0.

We will take up this situation in a more general setting. By an E_{0}-semigroup we mean a semigroup {α_{t} : t ≥ 0 } of self-adjoint normal endomorphisms of the algebra B(H) of all bounded operators on a Hilbert space H which preserves the identity (α_{t}(1) = 1 for every t ≥ 0) and is such that

t ↦< α_{t}(A)ξ, η >

is continuous for every ξ, η ∈ H and every A ∈ B(H). We emphasize that for the results discussed in the sequel, it is essential that the Hilbert space H should be separable. Naturally, one may speak of E_{0}-semigroups that act on type I factors M, and in this less concrete situation we require that the predual of M be separable. Two E_{0}-semigroups α (acting on M) and β (acting on N) are said to be conjugate if there is a ∗-isomorphism θ : M → N such that θ ∘α_{t} = β_{t} ∘θ for every t ≥ 0. One may also speak of E_{0}-semigroups that act on more general factors, but we shall not do so here.

The preceding discussion shows that one may obtain E_{0}-semigroups by starting with a strongly continuous one-parameter unitary group U = {U_{t} : t ∈ ℝ } acting on a Hilbert space H, finding somehow a type I subfactor M_{+} ⊆ B(H) which is invariant in the sense that

U_{t}M_{+}U_{t}^{∗} ⊆ M_{+}

for nonnegative t and then defining an E_{0}-semigroup α on M_{+} by setting

α_{t}(A) = U_{t}AU_{t}^{∗}, t ≥ 0, A ∈ M_{+}.

Notice that in this case there is a “complementary” E_{0}-semigroup. Indeed, the commutant M_{−} = M′_{+} of M_{+} is also a type I factor, and is invariant under the automorphisms A ↦ U_{t}AU_{t}^{∗} for negative t. Thus we can define a second E_{0}- semigroup β acting on M_{−} by way of

β_{t}(A) = U_{t}^{∗}AU_{t}, t ≥ 0, A ∈ M_{−}.

These remarks show that the E_{0}-semigroups that arise in this way from automorphism groups of a larger type I factor always occur in pairs. Several natural questions occur at this point. For example, does every abstract E_{0}-semigroup α which acts on a type I factor arise in this way from a one parameter group of automorphisms of a larger type I factor? If it does, then what is the relation between the “positive” and “negative” semigroups α and β? We will answer these and other questions presently. Let us first recall some known results pertaining to an analogous but much simpler situation.

Let U = {U_{t} : t ≥ 0 } be a strongly continuous one parameter semigroup of isometries acting on a Hilbert space H. Then H can be decomposed into a direct sum H_{1} ⊕ H_{2} giving rise to a corresponding decomposition

(1.2)  U = V ⊕ W

of U into a direct sum of two semigroups of isometries with the property that V is pure in that

⋂_{t≥0} V_{t}H_{1} = 0

and such that W_{t} is unitary for every t ≥ 0. This decomposition is unique. Moreover, the pure summand V is unitarily equivalent to a direct sum of a countable number n of copies of the shift semigroup S, which acts on the Hilbert space L^{2}(0, ∞) by way of

S_{t}f(x) = \{ \begin{matrix}f(x − t), & for x > t \\ 0, & for 0 < x ≤ t. \end{matrix}

The number n is also uniquely determined. Taken together, these results are often called the Wold decomposition. The summand H_{2} is defined by

H_{2} = ⋂_{t≥0} U_{t}H,

and of course H_{1} is the orthocomplement of H_{2}.

One of my first thoughts about E_{0}-semigroups was to speculate that there should be an effective decomposition resembling the Wold decomposition, and that this might lead to some kind of classification up to conjugacy. It quickly became clear that this idea was too naive. Since a discussion of this issue serves to make a significant point about E_{0}-semigroups, we offer the following comments.

Suppose that one is given an E_{0}-semigroup α = {α_{t} : t ≥ 0 } which acts on a type I factor M. We indicate the extent to which a decomposition resembling the Wold decomposition is valid for α. There are two natural von Neumann algebras associated with the action of α:

M_{∞} = ⋂_{t≥0} α_{t}(M),  and
M^{α} = {A ∈ M : α_{t}(A) = A, t ≥ 0 }.

M_{∞} is the “tail” von Neumann algebra, and M^{α} is the fixed algebra. We have M^{α} ⊆ M_{∞} ⊆ M, and the action of α_{t} on M_{∞} defines an automorphism of M_{∞} for every t ≥ 0. Thus the restriction of α to M_{∞} defines a one-parameter group of automorphisms of M_{∞}. This W^{∗}-dynamical system plays the role of the summand W in the Wold decomposition (1.2), and it is clearly a conjugacy invariant of the original E_{0}-semigroup α.

Let N = M′_{∞} denote the (relative) commutant of M_{∞}. Notice that since M is a factor of type I, we may treat relative commutants (in M) as if they were true commutants. Since α_{t}(M_{∞}) = M_{∞} for every t ≥ 0, it follows in a straightforward way that α_{t}(N) ⊆ N, t ≥ 0. Moreover, it is not hard to show that the restriction of α to N has trivial “tail” in the sense that

⋂_{t≥0} α_{t}(N) = center of N = center of M_{∞}.

In the important case where M_{∞} is a factor, then we have a decomposition resembling (1.2) to some extent, in that

(1.3)  M = N ∨ M_{∞}

is generated by a pair of mutually commuting factors which are commutants of each other, that α determines an automorphism group on M_{∞}, that α leaves N invariant and in fact defines a “pure” semigroup on N in the sense that

(1.4)  ⋂_{t≥0} α_{t}(N) = ℂ1.

If M_{∞} happens to be a factor of type I then (1.3) becomes a simple tensor product

(1.5)  M = N ⊗ M_{∞},

and if we define E_{0}-semigroups β on N and γ on M_{∞} by restricting α in the obvious way then we have a decomposition

(1.6)  α = β ⊗ γ.

Formula (1.6) appears quite analogous to the Wold decomposition (1.2).

Unfortunately, M_{∞} need not be a type I von Neumann algebra. Moreover, even if it were a type I factor, we do not know how to classify E_{0}-semigroups that satisfy (1.4). And if M_{∞} is a factor not of type I, then almost nothing is known about semigroups satisfying (1.4). Finally, it is known that there exist E_{0}-semigroups whose fixed algebra M^{α} is a factor of type II_{∞} or of type III (we will have more to say about this in sections 7 and 8).

These remarks lead one to the conclusion that E_{0}-semigroups are much too complex to hope for a useful classification up to conjugacy.

2. Cocycle perturbations.

Let α be an E_{0}-semigroup acting on B(H). A cocycle for α is a strongly continuous family of unitary operators U = {U_{t} : t ≥ 0 } satisfying

(2.1)  U_{s+t} = U_{s}α_{s}(U_{t}),  s, t ≥ 0.

Notice that (2.1) implies that U_{0} = 1. The condition also implies that the family of endomorphisms β = {β_{t} :≥ 0 } defined by

(2.2)  β_{t}(A) = U_{t}α_{t}(A)U_{t}^{∗},  t ≥ 0

is another E_{0}-semigroup acting on B(H). β is called a cocycle perturbation of α. Two E_{0}-semigroups are said to be cocycle conjugate if one of them is conjugate to a cocycle perturbation of the other. This is an important relation for the theory of E_{0}-semigroups, and we want to discuss the notion of cocycle perturbations in more detail.

Suppose that α and β are two E_{0}-semigroups which act on the same B(H), and that β is a cocycle perturbation of α as in (2.2):

β_{t}(A) = U_{t}α_{t}(A)U_{t}^{∗}, A ∈ B(H), t ≥ 0.

Notice that the unitary operators V_{t} = U_{t}^{∗}, t ≥ 0 define a cocycle for β and of course we have

α_{t}(A) = V_{t}β_{t}(A)V_{t}^{∗}.

Thus, this notion of cocycle perturbation defines an equivalence relation on the set of all E_{0}-semigroups that act on a fixed B(H).

Note too that this relation is analogous to Connes’ notion of exterior equivalence for one-parameter automorphism groups of a von Neumann algebra M [13]. In particular, Connes discovered that if one is given a pair of faithful normal weights of M, then the two associated modular automorphism groups are exterior equivalent, and this led to the establishment of an elegant classification theory for type III factors. Cocycle perturbations are of fundamental importance for the classification of E_{0}-semigroups as well, but for reasons that involve new phenomena that are unique to the theory of E_{0}-semigroups (see Theorem B below).

Remark 2.3. The cohomology of cocycle perturbations. Suppose first that α and β are conjugate E_{0}-semigroups which act on the same B(H), say θ ∘ β_{t} = α_{t} ∘ θ where θ is a ∗-automorphism of B(H). We may find a unitary operator W that implements θ in the sense that θ(A) = W^{∗}AW, A ∈ B(H). It follows that

β_{t}(A) = U_{t}α_{t}(A)U_{t}^{∗}, t ≥ 0 A ∈ B(H),

where U is the α-cocycle defined by U_{t} = W α_{t}(W)^{∗}. An α-cocycle of this form is called exact. Conversely, if β is a perturbation of α by an exact cocycle, then α and β are conjugate. This remark explains why the problem of classifying E_{0}- semigroups to conjugacy is difficult: it is the same as the problem of computing a very subtle noncommutative cohomology group.

Remark 2.4. Perturbations of the infinitesimal generator. We want to point out a useful interpretation of cocycle conjugacy in terms of infinitesimal generators. Suppose that α acts on B(H), and let δ be the generator of α. For an operator A in the domain of δ, δ(A) is defined as the limit in the strong operator topology

δ(A) = lim_{t→0+} t^{−1}(α_{t}(A) − A).

The domain of δ is a unital ∗-subalgebra D of B(H) and δ is a self-adjoint derivation of D into B(H). Let B be any bounded self adjoint operator in B(H) and put

δ′(A) = δ(A) + i(BA − AB).

δ′ is another unbounded self-adjoint derivation having the same domain D, and it can be shown that it is the generator of a second E_{0}-semigroup β which acts on B(H). In fact, β is a cocycle perturbation of α and the cocycle U relating β to α as in (2.2) can be defined as the global solution of the linear differential equation

\frac{d}{dt}U(t) = iU(t)α_{t}(B), t ≥ 0,

with the initial condition U(0) = 1. It is easy to see from the properties of this differential equation and the fact that B is bounded that the cocycle U is norm continuous:

lim_{t→t_{0}} ∥U(t) − U(t_{0})∥ = 0,

for every t_{0} ≥ 0. Now in general, cocycles need not be norm continuous, and the generators of cocycle perturbations of α cannot be obtained by perturbing the generator of α by bounded derivations. Nevertheless, it is useful to think of the cocycle perturbations of an E_{0}-semigroup as having been obtained by perturbing its generator by an “unbounded” derivation. While this interpretation is merely a heuristic conceptual device, it can sometimes be made precise. In any event, one should consider the definition of cocycle perturbation as a precise formulation of this idea.

3. Numerical index.

The preceding discussion shows that one should look for cocycle conjugacy invariants. We now describe a numerical index invariant which is appropriately thought of as a quantized form of the Fredholm index of certain differential operators (or of the operator semigroups that they generate). It is defined as follows.

Let α = {α_{t} : t ≥ 0 } be an E_{0}-semigroup acting on B(H). A unit for α is a strongly continuous semigroup of bounded operators {U_{t} : t ≥ 0} on H which intertwines α in the sense that

α_{t}(A)U_{t} = U_{t}A, t ≥ 0, A ∈ B(H),

and is not the zero semigroup U_{t} = 0. U_{α} will denote the set of all units of α. Notice that if U and V belong to U_{α} then for each t ≥ 0 V_{t}^{∗}U_{t} is a bounded operator which, by 3.1, commutes with every bounded operator on H, and hence must be a scalar multiple of the identity

V_{t}^{∗}U_{t} = f(t)1, t ≥ 0.

f is a continuous complex-valued funciton satisfying f(0) = 1, and one easily verifies that f(s + t) = f(s)f(t) for all nonnegative s, t. Hence there is a unique complex number c(U, V) satisfying

f(t) = e^{c(U,V)t},

for every t ≥ 0. The bivariate function

c : U_{α} × U_{α} → ℂ

is called the covariance function of α. The covariance function is easily seen to be conditionally positive definite. Thus one may use the pair (U_{α}, c) in a natural way to construct a Hilbert space H_{α}. It can be shown that H_{α} is separable. We define

d_{∗}(α) = dim H_{α}.

Thus the possible values of d_{∗}(α) are 0, 1, 2, …, ℵ_{0}. Significantly, there are E_{0}- semigroups α for which U_{α} = ∅ [18], and in this case it is convenient for the arithmetic of the index to define d_{∗}(α) = 2^{ℵ_{0}}. See [2] for more detail.

The basic property of this index is that it behaves well under the formation of tensor products [6]. If α and β are two E_{0}-semigroups acting, respectively, on B(H) and B(K) then it is easy to see that there is a unique E_{0}-semigroup α ⊗ β, acting on B(H ⊗ K), which satisfies

(α ⊗ β)_{t} : A ⊗ B ↦ α_{t}(A) ⊗ β_{t}(B)

for every t ≥ 0, A ∈ B(H), B ∈ B(K).

Theorem A.

d_{∗}(α ⊗ β) = d_{∗}(α) + d_{∗}(β).

Notice that d_{∗}(α) is invariant under cocycle conjugacy, or what is in substance the same, that d_{∗}(α) is stable under cocycle perturbations of α. That is easily seen as follows. Let W = {W_{t} : t ≥ 0 } be any cocycle for α and consider the associated perturbation β

β_{t}(A) = W_{t}α_{t}(A)W_{t}^{∗}, t ≥ 0, A ∈ B(H).

One may check that if U = {U_{t} : t ≥ 0 } is a unit for α and if we define

Ũ_{t} = W_{t}U_{t}, t ≥ 0,

then Ũ is a unit for β. Moreover, U ↦ Ũ is a bijection of U_{α} onto U_{β} which carries one covariance function into the other. Thus the Hilbert spaces H_{α} and H_{β} are isomorphic and it follows that d_{∗}(α) = d_{∗}(β).

This definition of numerical index is equivalent to the one given in [3], and differs significantly from Powers’ earlier definition of numerical index [18]. In the latter the index was defined as the multiplicity of a certain representation of a C^{∗}- algebra associated with the infinitesimal generator of α. Since the representation depended on making a particular choice of a unit, it was not clear that this index was unambiguously defined. Later, Powers and Robinson [22] gave another definition of index which was obviously well-defined, but which took values in an abstract set of equivalence classes rather than in the nonnegative integers. Recently, Powers and Price [23] have shown that d_{∗}(α) actually agrees with Powers’ “infinitesimal” definition in all cases. In particular, it is now clear that Powers’ original definition of the numerical index was unambiguous.

4. Continuous tensor product systems.

We now want to emphasize a fundamental relationship between E_{0}-semigroups and continuous tensor products of Hilbert spaces. Indeed, one could argue that up to cocycle conjugacy, the theory of E_{0}-semigroups is the theory of continuous tensor products of Hilbert spaces.

Heuristically, a product system is a measurable family of Hilbert spaces E = {E_{t} : t > 0 } which behaves as if each E_{t} were a continuous tensor product

E_{t} = ⨂_{0<s<t} H_{s}, H_{s} = H

of copies of a single separable Hilbert space H. While this heuristic picture is often useful, one must be careful not to push it too far. Indeed, we will see that this picture is basically correct for the simpler examples of product systems, but that there are other examples with the remarkable property that the “germ” H fails to exist. Rather than reiterate the details of the precise definition here, we illustrate the essentials of the structure of product systems in the discrete case, where the positive real line is replaced with the discrete set ℕ = {1, 2, … } of positive integers. Then we will indicate briefly how to change the axoims to pass from ℕ to ℝ^{+}. Full details can be found in [2].

Let H be a separable Hilbert space. For every n = 1, 2 … let E(n) be the full tensor product of n copies of H:

E(n) = \underbrace{H ⊗ H ⊗ ⋯ ⊗ H}_{n times} .

We may organize these spaces into a family of Hilbert spaces p : E → ℕ over ℕ by setting

E = {(t, ξ) : t ∈ ℕ, ξ ∈ E(t) },

with projection p(t, ξ) = t. We introduce an associative multiplication on the structure E by making use of the tensor product

(s, ξ) · (t, η) = (s + t, ξ ⊗ η),

ξ ∈ E(s), η ∈ E(t). This multiplication is bilinear on fibers, and has the two additional properties

(4.1)  E(s + t) = \overline{span} E(s) · E(t),  s, t ∈ ℕ
(4.2)  < ux, vy > =< u, v >< x, y >, ∀u, v ∈ E(s), x, y ∈ E(t).

Notice that the Hilbert space associated with the sections of p : E → ℕ is the direct sum

∑_{t∈ℕ} E(t) = ∑_{n=1}^{∞} H^{⊗n},

essentially the full Fock space over the one-particle space H.

A unit is a section n ∈ ℕ ↦ u_{n} ∈ E(n) satisfying

u_{m+n} = u_{m}u_{n},

and which is not the zero section. The most general unit has the form

u_{n} = \underbrace{x ⊗ x ⊗ ⋯ ⊗ x}_{n times}

n ≥ 1, where x is a nonzero element of the one-particle space H.

Fix n = 1, 2 … and let u be a vector in E_{n}. u is called decomposable if for every k = 1, 2, …, n − 1 there are vectors v_{k} ∈ E(k), w_{k} ∈ E(n − k) such that

u = v_{k}w_{k}.

Notice that the most general decomposable vector in E(n) is an elementary tensor of the form

u = x_{1} ⊗ x_{2} ⊗ ⋯ ⊗ x_{n},

where x_{k} ∈ H for k = 1, 2, …, n.

A product system is a similar structure, except that it is associated with the space of positive reals rather than ℕ. More precisely, a product system is a family of separable Hilbert spaces over the semi-infinite interval (0, ∞)

p : E → (0, ∞)

which admits an associative multiplication that is bilinear on fiber spaces and has properties analogous to properties corresponding to (4.1) and (4.2). Additionally, E is endowed with a standard Borel structure which is compatible in the natural way with the other structures of E, and with the further property that there should be a separable Hilbert space H such that

(4.3)  E ≅ (0, ∞) × H,

where ≅ denotes an isomorphism of measurable families of Hilbert spaces. Condition (4.3) is nontrivial, and is the property in this category that corresponds to local triviality of Hermitian vector bundles. There is a natural notion of isomorphism for the category of product systems, which we will not write down explicitly here (see [2]). We may also define units of E and decomposable vectors of the fiber spaces E_{t} in a way analogous to the above. For example, a unit is a measurable cross section

u : t ∈ (0, ∞) ↦ u(t) ∈ E_{t}

satisfying u(s + t) = u(s)u(t) for all s, t > 0, and which is not the zero section.

One might expect that it should be possible to write down a comprehensive list of (continuous) product systems as we have done above for their discrete analogues. In that case there is, up to isomorphism, exactly one “discrete” product system for every integer d = 1, 2, …, ℵ_{0}. d can be taken to be the dimension of the one-particle space E_{1}. In the continuous case, however, nothing like that is true. While there is a family of “natural” examples parameterized by the values d = 1, 2, …, ℵ_{0}, there are many others as well. The problem of classifying general product systems is an unsolved problem which, as we will see, is of central importance in the theory of E_{0}-semigroups.

Every E_{0}-semigroup α = {α_{t} : t ≥ 0 } gives rise to a product system E_{α} in the following way. Suppose α acts on B(H). For every t > 0, let E_{α}(t) be the intertwining space

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

E_{α}(t) is obviously a complex vector space, and in fact it is a Hilbert space. Indeed, if S, T ∈ E_{α}(t) then because of the intertwining property it follows that T^{∗}S commutes with every operator A ∈ B(H). Hence there is a unique complex number < S, T > such that

T^{∗}S =< S, T > 1.

<, > is an inner product on E_{α}(t) which makes it into a Hilbert space. Thus we have a family of Hilbert spaces p : E_{α} → (0, ∞) defined by

E_{α} = {(t, ξ) : t > 0, ξ ∈ E_{α}(t) }

where p(t, ξ) = t. If we use operator multiplication to define multiplication in E_{α} by

(s, S) · (t, T) = (s + t, ST),

then it is not hard to establish the properties (4.1) and (4.2). E_{α} inherits a natural standard Borel structure as a subspace of (0, ∞) × B(H), where B(H) is endowed with the Borel structure generated by its weak^{∗} topology. Finally, it is a nontrivial fact that property (4.3) is valid as well [2]. Thus E_{α} is a product system. The importance of product systems in this context derives from the following result from [2].

Theorem B. α and β are cocycle conjugate iff E_{α} and E_{β} are isomorphic product systems.

In order to illustrate what lies behind Theorem B, let us consider the case in which α and β both act on B(H) and β is a cocycle perturbation of α:

β_{t}(A) = U_{t}α_{t}(A)U_{t}^{∗}, t ≥ 0, A ∈ B(H),

where U = {U_{t} : t ≥ 0 } is an α-cocycle. In this case it is quite easy to exhibit an isomorphism of product systems θ : E_{α} → E_{β}. Indeed, if we fix t > 0 and choose T ∈ E_{α}(t), then one may verify directly that U_{t}T belongs to E_{β}(t). Thus we can define a unitary operator θ_{t} : E_{α}(t) → E_{β}(t) by θ_{t}(T) = U_{t}T; θ is defined as the total map. θ is a (measurable) bijection which is unitary on fibers, hence it is an isomorphism of families of Hilbert spaces. The cocycle property implies that for s, t > 0 and S ∈ E_{α}(s), T ∈ E_{α}(t) we have

θ_{s}(S)θ_{t}(T) = U_{s}SU_{t}T = U_{s}α_{s}(U_{t})ST
= U_{s+t}ST = θ_{s+t}(ST).

Thus θ preserves multiplication, and hence it is an isomorphism of product systems.

To prove the converse direction (still assuming that α and β act on the same B(H)), one basically has to start with an isomorphism θ : E_{α} → E_{β} and show that θ is associated with a unitary α-cocycle U as above. This is technically more difficult, but the basic idea is similar (see [2], Theorem 3.18).

Theorem B implies that the problem of classifying E_{0}-semigroups up to cocycle conjugacy is equivalent to that of classifying certain product systems...namely, those product systems that can be associated with an E_{0}-semigroup as above. It is a basic result in our approach to E_{0}-semigroups that every product system arises in this way.

Theorem C. For every product system E there is an E_{0}-semigroup α such that E is isomorphic to E_{α}.

The proof of Theorem C is very indirect [5], and makes essential use of the spectral C^{∗}-algebras discussed in sections 7 and 8.

We want to point out that there is a general notion of dimension of an abstract product system that generalizes the numerical index of E_{0}-semigroups. This dimension function takes values in the set {0, 1, 2, …, ℵ_{0}, 2^{ℵ_{0}}}, and corresponding to Theorem A it obeys

dim(E ⊗ F) = dim E + dim F

where ⊗ denotes the natural tensor product in the category of product systems. The relation of d_{∗} to dim is given by the expected formula

d_{∗}(α) = dim(E_{α}).

It follows that, in order to classify E_{0}-semigroups up to cocycle conjugacy, one should seek to determine the structure of product systems. In particular, we may consider the set Σ of all isomorphism classes of product systems. The class of a product system E will be denoted [E]. There is a natural “addition” in Σ, defined by the natural tensor product operation

[E] + [F] = [E ⊗ F],

which makes Σ into an abelian semigroup. There is a neutral element, which arises from the trivial product system Z. Z is defined as the trivial family of one-dimensional Hilbert spaces

Z = {(t, z) : t > 0, z ∈ ℂ }

where the inner product in ℂ is the usual one < z, w >= z\overline{w}. The multiplication in Z is given by

(s, z) · (t, w) = (s + t, zw).

It can be seen that if α is an E_{0}-semigroup which is trivial in the sense that each α_{t} is an automorphism, then E_{α} is isomorphic to Z. Moreover, it is also a fact that there are no nontrivial line bundles in Σ: every product system E with onedimensional fiber spaces E_{t}, t > 0, is isomorphic to Z [6]. One can verify directly that for every [E] ∈ Σ one has

[E ⊗ Z] = [Z ⊗ E] = [E].

Therefore [Z] functions as an additive zero for Σ.

There is also a natural involution in Σ, defined by [E] ↦ [E^{o}], where E^{o} is the product system opposite to E. The structure of E^{o} is identical to that of E except that multiplication is reversed. With this involution, Σ becomes an abelian involutive semigroup with a zero element. The problem of classifying E_{0}-semigroups up to cocycle conjugacy becomes the probem of determining the structure of the involutive semigroup Σ.

At this point, we are not even certain of the cardinality of Σ! It is expected that Σ is uncountable, but this has not been proved. Notice for example that by the more general version of Theorem A alluded to above, the dimension function defines a homomorphism of Σ into the additive semigroup of extended nonnegative integers {0, 1, 2, …, ℵ_{0}, 2^{ℵ_{0}}}. Little is known about the quotient structure in Σ defined by this homomorphism. For example, by recent work of Powers in which a new family of E_{0}-semigroups is constructed [21], we now know that for each k = 1, 2, …, ℵ_{0} there are infinitely many elements x of Σ that satisfy

dim(x) = k.

But it is still not known if there are distinct elements x, y ∈ Σ satisfying dim(x) = dim(y) = 0. Equivalently, is there a nontrivial E_{0}-semigroup α with the property that there is a nonzero unit U = {U_{t} : t ≥ 0 } and such that every other unit V is related to U by a relation of the form

V_{t} = e^{iλt}U_{t}, t ≥ 0

where λ is a complex number?

Finally, let us return to some questions raised in section 1 concerning the problem of extending E_{0}-semigroups to automorphism groups acting on a larger type I factor. Suppose that we are given a pair of E_{0}-semigroup s α, β acting respectively on B(H) and B(K). We are interested in obtaining conditions on the pair α, β which imply that there is a one-parameter group of unitaries W = {W_{t} :∈ ℝ} acting on the tensor product H ⊗ K whose associated automorphism group γ_{t}(C) = W_{t}CW_{t}^{∗} satisfies

γ_{t}(A ⊗ 1) = α_{t}(A) ⊗ 1, for t ≥ 0
γ_{t}(1 ⊗ B) = 1 ⊗ β_{−t}(B), for t ≤ 0.

In case such a group exists, then α and β are said to be paired. This relation was introduced by Powers and Robinson in [22] as an intermediate step in their definition of another index. The Powers-Robinson index is an equivalence relation defined in the class of all E_{0}-semigroups; α and β are said to have the same index if there is a third E_{0}-semigroup σ with the property that α can be paired with σ and σ can be paired with β.

Using the theory of product systems one can determine the precise nature of this pairing, and thus give a more concrete form to the Powers-Robinson index. The details are as follows. It is not hard to show that α and β are paired iff their product systems E_{α} and E_{β} are anti isomorphic. Moreover, with any particular anti isomorphism θ : E_{α} → E_{β} one can write down a specific one parameter unitary group W acting on H ⊗K which simultaneously extends α and β in the above sense (the details can be found in [2, pp 27–28]).

Carrying this one step further, we can answer a question posed in section 1 which asks how to describe the possible ways of extending an E_{0}-semigroup to a larger type I factor. More precisely, starting with a particular E_{0}-semigroup α we seek to describe all possible ways of finding a one-parameter unitary group W = {W_{t} : t ∈ ℝ} acting on some other Hilbert space K and a type I subfactor M ⊆ B(K) with the property that

W_{t}MW_{t}^{∗} ⊆ M, for t ≥ 0

and such that α is conjugate to the restriction of adW_{t}, t ≥ 0 to M. The preceding remarks show that one should begin by considering the product system E_{α}^{o} opposite to E_{α}. Notice that by Theorem C, there exist E_{0}-semigroups whose product systems are isomorphic to E_{α}^{o}. Moreover, the set of all possible extensions of α is described by the set of all anti isomorphisms of E_{α} to E_{α}^{o}. In turn, these are obtained by composing the natural anti isomorphism of E_{α} to E_{α}^{o} with an arbitrary automorphism of E_{α} itself. Thus the set of all possible extensions of α is parameterized by the group of all automorphisms of the product system E_{α}. This group is computed explicitly for the “standard” examples in ther last section of [2]; its structure in the case of general product systems remains mysterious.

In particular, it follows from these remarks that two E_{0}-semigroups have the same index in the sense of Powers-Robinson iff their product systems determine the same element of the semigroup Σ. Thus, this discussion also gives a somewhat more concrete description of the Powers-Robinson index: it is now identified with this Σ-valued index map

α → [E_{α}] ∈ Σ.

5. CCR flows.

We have already remarked that there is a sequence of standard E_{0}-semigroups, and corresponding to them a sequence of standard product systems. These have been described in detail in [2] and [7]. The purpose of this section is to give a brief description of these standard examples and to describe recent results on the problem of characterizing E_{0}-semigroups that are cocycle conjugate to a standard one.

It is useful to think of this construction as a functor related to second quantization; that interpretation makes explicit the precise sense in which the index d_{∗}(α) of an E_{0}-semigroup is a quantized form of the Fredholm index of certain differential operators. A fuller discussion of these issues can be found in [7].

Consider the category S whose objects are semigroups of isometries U = {U_{t} : t ≥ 0} each of which acts on a separable Hilbert space H_{U}. hom(U, V) consists of unitary operators W : H_{U} → H_{V} which intertwine U and V:

W U_{t} = V_{t}W, t ≥ 0.

This category admits a natural direct sum operation ⊕, in which U ⊕ V is the semigroup of isometries on H_{U} ⊕ H_{V} defined by

(U ⊕ V)_{t} = U_{t} ⊕ V_{t}, t ≥ 0.

Every semigroup of isometries U decomposes uniquely into a direct sum

(5.1)  U = V ⊕ W

where W is a semigroup of unitary operators and where V is pure in the sense that

⋂_{t>0} V_{t}H_{V} = 0.

Moreover, every pure semigroup of isometries is isomorphic to a direct sum of a countable number d of copies of the simple shift semigroup S which acts on L^{2}(0, ∞) by way of

S_{t}f(x) = \{ \begin{matrix}f(x − t), & for x > t \\ 0, & for 0 < x ≤ t. \end{matrix}

The number d of copies of S is uniquely determined by V, and is called the index of V. We remark that there are other ways to define the index of V, involving the deficiency spaces of the infinitesimal generator of V. But the definition we have given is the quickest. All of this information about the decomposition (5.1) is often referred to as the Wold decomposition.

The index obeys the expected law of addition

ind(U_{1} ⊕ U_{2}) = ind(U_{1}) + ind(U_{2}),

and it can take on any of the values 0, 1, 2, …, ℵ_{0}. Notice that there is also a tensor product operation defined on S, but it has terrible arithmetic properties with respect to the index. For example, if U_{1} and U_{2} have index 1 then U_{1} ⊗ U_{2} has index ℵ_{0}. Thus we must consider S as a category with a single operation ⊕.

Let E be the category whose objects are E_{0}-semigroups and whose maps are conjugacies. Thus, if α and β are E_{0}-semigroups acting respectively on B(H_{α}) and B(H_{β}), then hom(α, β) consists of ∗-isomorphisms θ : B(H_{α}) → B(H_{β}) satisfying

θ(α_{t}(A)) = β_{t}(θ(A))

for every t ≥ 0, A ∈ B(H_{α}). The natural operation in E is the tensor product of E_{0}-semigroups that has already been defined in section 3.

One might well ask if there is a direct sum operation in E. Assuming that there were such an operation, one would expect α ⊕ β to be an E_{0}-semigroup acting on B(H_{α} ⊕ H_{β}). By replacing β with a conjugate copy of itself if necessary, we may assume that H_{α} = H_{β} = H. Then B(H ⊕ H) consists of 2 × 2 matrices over B(H). One would expect at least that α ⊕ β should restrict to α and β on the appropriate summands, that is α ⊕ β should also have the property

(5.2)  (α ⊕ β)_{t} : ( \begin{matrix}A & 0 \\ 0 & B \end{matrix} ) → ( \begin{matrix}α_{t}(A) & 0 \\ 0 & β_{t}(B) \end{matrix} ) ,

for all t ≥ 0 and all A, B ∈ B(H). However, if there were an E_{0}-semigroup on B(H ⊕ H) which satisfied (5.2), then one could show that α and β must in fact be cocycle perturbations of each other. This argument is a variation of Connes’ elegant observation about exterior equivalence of modular automorphism groups (the idea can be found in Lemma 8.11.2 of [17]). In particular, if α and β are not cocycle conjugate then it is impossible to make a reasonable definition of α ⊕ β. One might summarize this state of affairs as follows: the only appropriate operation in the category S is the direct sum and the only possible operation in the category E is the tensor product.

Finally, the index is defined on the objects of E and because of Theorem A we have

d_{∗}(α ⊗ β) = d_{∗}(α) + d_{∗}(β).

We now describe a construction which can be considered a functor from S to E. This is (Boson) quantization, and it is also a form of exponentiation in that direct sums carry over to tensor products. The details are as follows.

Let H be a Hilbert space. We will write H^{n} for the symmetric tensor product of n copies of H if n ≥ 1; H^{0} is defined as ℂ. Let

e^{H} = ∑_{n=0}^{∞} H^{n}

be the symmetric Fock space over H. The natural exponential map exp : H → e^{H} is defined by

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

e^{H} is spanned by the vectors of the form exp(ξ), ξ ∈ H, and we have

< exp(ξ), exp(η) >= e^{<ξ,η>}.

For every ξ ∈ H there is a unique unitary operator W (ξ) on e^{H} which satisfies

W (ξ) exp(η) = e^{− \frac{1}{2} ∥ξ∥^{2}−<η,ξ>}exp(η + ξ).

W is strongly continuous, obeys Weyl’s form of the canonical commutation relations

W (ξ)W (η) = e^{iIm<ξ,η>}W (ξ + η),

and W (H) is an irreducible set of operators on e^{H}.

Now let U ∈ S. We define an E_{0}-semigroup α^{U} on B(e^{H}) as follows. Fix t ≥ 0. Because of the irreducibility of W, it is not hard to verify that there is a unique normal endomorphism α_{t}^{U} of B(e^{H}) satisfying

α_{t}^{U}(W (ξ)) = W (U_{t}ξ), ξ ∈ H_{U}.

α^{U} = {α_{t}^{U} : t ≥ 0 } is an E_{0}-semigroup. Because of the natural identification

e^{H_{1}⊕H_{2}} = e^{H_{1}} ⊗ e^{H_{2}}

we have a natural identificaiton

(5.3)  α^{U⊕V} = α^{U} ⊗ α^{V}

for every U, V ∈ S. With these observations in hand, one can establish the functoriality of the map U → α^{U} (see [7] for more detail).

Let U ∈ S. Applying (5.3) and the Wold decomposition (5.1), we find that α^{U} decomposes into a tensor product

α^{U} = β ⊗ γ

where γ is a trivial E_{0}-semigroup (i.e., each γ_{t} is an automorphism) and where β is conjugate to an E_{0}-semigroup of the form α^{d·S} where d · S is a direct sum of d copies of the simple shift semigroup S acting on L^{2}(0, ∞). An E_{0}-semigroup such as β is called a CCR flow of index d. This terminology is justified by the following index theorem [2],[7].

Theorem D.

d_{∗}(α^{U}) = index(U).

The proof of Theorem D involves some work: one must find all the units of α^{U} and compute the associated covariance function in order to calculate the dimension of its associated Hilbert space. It follows immediately from Theorem B that if α is any E_{0}-semigroup and γ is a trivial E_{0}-semigroup, then α⊗γ is cocycle conjugate to α (actually, it is not hard to prove the latter directly) . If we collect this observation together with Theorem D then we are led to the conclusion that for any U, V ∈ S, α^{U} is cocycle conjugate to α^{V} iff α^{U} and α^{V} have the same index. Thus the numerical index is a complete cocycle conjugacy invariant for these examples.

We also point out that there is a corresponding construction of “standard” examples of product systems which parallels what we have done for E_{0}-semigroups, and we refer the reader to [2] for the details.

Finally, one may use the Fermionic Fock space to construct standard examples of E_{0}-semigroups having index 1, 2 …, ℵ_{0} in a way that is roughly parallel to what we have done above (though we point out that the construction of the corresponding product systems is not so explicit in the Fermionic setting). The E_{0}-semigroups obtained from either the Bosonic or the Fermionic construction turn out to be conjugate, provided that their numerical index is the same.

6. Classification results.

In this section we will describe two characterizations of product sytems which are isomorphic to one of the standard product systems. These results provide a classification of E_{0}-semigroups which have sufficiently many units, or which have enough decomposable operators.

Let α = {α_{t} : t ≥ 0 } be an E_{0}-semigroup acting on B(H), and fix t > 0. Consider the set S_{t} of all operators which can be decomposed into a finite product of the form

(6.1)  T = U_{1}(t_{1})U_{2}(t_{2}) … U_{n}(t_{n}),

where the U_{1}, U_{2}, …, U_{n} are units for α, where t_{1}, t_{2}, …, t_{n} are positive real numbers summing to t, and where n is an arbitrary positive integer. Because t_{1} + t_{2} + ⋯ + t_{n} = t it is clear that S_{t} ⊆ E_{α}(t) for every t > 0. We say that α is completely spatial if there is a t > 0 such that

(6.2)  H = \overline{span}\{T ξ : T ∈ S_{t}, ξ ∈ H \}.

It is easy to see that if (6.2) is true for some particular positive t, then it is true for every positive t.

It was proved in ([2] section 7) that every completely spatial E_{0}-semigroup is cocycle conjugate to a CCR flow. It follows that completely spatial E_{0}-semigroups are classified by their numerical index. This is proved at the level of product systems, using Theorem B.

That result has recently been extended significantly [9]. The extended version does not assume the existence of units, and is formulated as follows. Fix t > 0. An operator T ∈ E_{α}(t) is called decomposable if, for every 0 < s < t there are operators A_{s} ∈ E_{α}(s), B_{s} ∈ E_{α}(t − s) such that

T = A_{s}B_{s}.

We write D_{t} for the set of all decomposable operators in E_{α}(t). Any operator of the form (6.1) is decomposable because of the semigroups property of each U_{k}, and therefore D_{t} contains S_{t}. In fact, it is not hard to see that the following conditions are equivalent

(6.3.1)  E_{α}(t) = \overline{span} D_{t}
(6.3.2)  H = \overline{span}\{T ξ : T ∈ D_{t}, ξ ∈ H\},

and that if the conditions (6.3) are satisfied for some particular t then they are satisfied for every positive t. We remark that one uses the Hilbert space topology on E_{α}(t) in (6.3.1); hence it is apparent that the conditions (6.3) depend only on the structure of the product system E_{α} associated with α. The main result of [9] is

Theorem E. Every E_{0}-semigroup satisfying the conditions (6.3) is cocycle conjugate to a CCR flow.

Utilizing an ingenious construction in [19], Powers showed that there are E_{0}- semigroups which possess no units whatsoever. In a recent paper [21] he also proved that there are E_{0}-semigroups which have units but which are not cocycle conjugate to a CCR flow. It follows that there are product systems which a) have no units, and others which b) have units but not enough units to generate the product system.

We believe that it should be possible to give a more direct consturction of product systems with these properties. Unfortunately, we do not yet know how to do this. There are examples of product systems that arise naturally in probability theory (see [2], pp 14–16). Some of these examples do not appear to contain enough units. However, in all such cases we have studied we eventually found many units that were not initially obvious. What is immediately obvious in these probabilistic examples is that the product systems are generated by their decomposable vectors. Theorem E tells us that such product systems must be standard ones. In particular, any attempt to construct nonstandard product systems from “decomposable” sets must fail.

7. Spectral invariant.

Theorem C above asserts that every abstract product system is associated with an E_{0}-semigroup. This result is analogous to the fact that every locally compact group G has a faithful unitary representation on a Hilbert space. The proof of the latter assertion about groups follows from an analysis of the properties of the group C^{∗}-algebra C^{∗}(G), together with the Gelfand-Neumark theorem. In that result, C^{∗}(G) functions as the “spectrum” of the group G.

In this section we show how, starting with a product system E, one can construct a spectral C^{∗}-algebra C^{∗}(E). Results like Theorem C are obtained by exploiting the properties of C^{∗}(E). More generally, C^{∗}(E) provides a “topological” invariant that is important for understanding the nature of product systems and their associated E_{0}-semigroups.

Let p : E → (0, ∞) be a product system, and let us write E(t) = p^{−1}(t) for the Hilbert space over t > 0. We form the Hilbert space of L^{2} sections

L^{2}(E) = ∫_{(0,∞)}^{⊕} E(t) dt .

The inner product in L^{2}(E) is the natural one

< ξ, η >= ∫_{0}^{∞} < ξ(t), η(t) > dt .

Let f ∈ L^{1}(E) be an integrable section. Using the multiplication in E we see that for every ξ ∈ L^{2}(E) and every 0 < t < x,

f(t)ξ(x − t) ∈ E(x),

and hence we can define a measurable section f ∗ ξ by

f ∗ ξ(x) = ∫_{0}^{x} f(t)ξ(x − t) dt .

For fixed f ∈ L^{1}(E), left convolution by f, ξ ↦ f ∗ ξ, defines a bounded linear operator on L^{2}(E) of norm at most ∥f∥_{1}. This operator is denoted l_{f}. A straightforward computation shows that for any two functions f, g ∈ L^{1}(E), there are functions h_{1}, h_{2} ∈ L^{1}(E) such that

l_{f}^{∗}l_{g} = l_{h_{1}} + l_{h_{2}}^{∗}.

It follows that the linear span of all products of the form l_{f}l_{g}^{∗} is a self-adjoint subalgebra of B(L^{2}(E)). C^{∗}(E) is defined as the norm-closure of this algebra

(7.1)  C^{∗}(E) = \overline{span}\{l_{f}l_{g}^{∗} : f, g ∈ L^{1}(E) \}.

The funcamental property of C^{∗}(E) is that its representations correspond to all possible E_{0}-semigroups α for which E_{α} is isomorphic to E. This is a key result in the theory and we want to state it precisely. It is convenient to slightly generalize the notion of E_{0}-semigroup. By an e_{0}-semigroup we mean a semigroup α = {α_{t} : t ≥ 0} of normal ∗-endomorphisms of B(H) that satisfies all of the conditions of an E_{0}- semigroup except that α_{t}(1) is not required to be 1. Thus, for an e_{0}-semigroup α,

P_{t} = α_{t}(1)

defines a strongly continuous family of projections which decreases as t → ∞. The limit

P_{∞} = lim_{t→∞} α_{t}(1)

is an α-invariant projection which induces a decomposition of the underlying Hilbert space

H = H_{∞} ⊕ H_{0},

where H_{∞} and H_{0} are, respectively, the ranges of the projections P_{∞} and 1 − P_{∞}. The restriction of α to B(H_{∞}) is an E_{0}-semigroup, and the restriction α^{0} of α to B(H_{0}) is an e_{0}-semigroup whose limiting projection is zero:

(7.2)  lim_{t→∞} α_{t}^{0}(1) = 0.

To this extent the study of e_{0}-semigroups reduces to the study of E_{0}-semigroups and the extreme case of e_{0}-semigroups satisfying (7.2).

We need to relate the representations of C^{∗}(E) more directly to E. By a representation of E we mean a measurable mapping ϕ : E → B(H) which restricts to a linear map on each fiber E(t), t > 0, which preserves multiplication in that ϕ(u)ϕ(v) = ϕ(uv) for all u, v ∈ E, and which obeys the following partial “commutation relation” on each fiber:

(7.3)  ϕ(v)^{∗}ϕ(u) =< u, v > 1,  u, v ∈ E(t).

If ϕ : E → B(H) is an arbitrary representation then we can define an e_{0}-semigroup α = {α_{t} : t > 0} which acts on B(H) as follows. For each positive t, choose an orthonormal basis {e_{1}(t), e_{2}(t), … } for E(t) and put

(7.4)  α_{t}(A) = ∑_{n=1}^{∞} ϕ(e_{n}(t))Aϕ(e_{n}(t))^{∗},

A ∈ B(H). We define α_{0}(A) = A. The left side is independent of the particular choice of basis {e_{n}(t)}, and it is true (though nontrivial) that α is an e_{0}-semigroup whose canonical product system is isomorphic to E. Indeed,

E_{α}(t) = {ϕ(v) : v ∈ E(t)}

and ϕ itself implements the stated isomorphism of E and Eα. These things are proved in [3].

More significantly, any representation ϕ : E → B(H) determines a unique representation π : C^{∗}(E) → B(H) by way of

(7.5)  π(l_{f}l_{g}^{∗}) = ϕ(f)ϕ(g)^{∗},  f, g ∈ L^{1}(E)

where for f ∈ L^{1}(E), ϕ(f) denotes the operator integral

ϕ(f) = ∫_{0}^{∞} ϕ(f(x)) dx.

The key property of C^{∗}(E) is that this association ϕ → π is in fact a bijection [3]:

Theorem F. The nondegenerate separable representations of C^{∗}(E) correspond bijectively with all e_{0}-semigroups α for which E_{α} is isomorphic to E.

Because of Theorem F, we are let to examine the structure of C^{∗}(E), and attempt to describe its state space in terms that are as explicit as possible. The remainder of this paper is devoted to a discussion of progress on these issues.

The principal result of [3] is that C^{∗}(E) is a simple C^{∗}-algebra in most (and perhaps all) cases. More precisely,

Theorem G. For every product system E, C^{∗}(E) is a unitless nuclear C^{∗}-algebra. If E possesses a nonzero unit, then C^{∗}(E) has no closed nontrivial ideals.

In particular, the C^{∗}-algebras C^{∗}(E_{n}), n = 1, 2, …, ℵ_{0} associated with the standard examples E_{n} (i.e., the product systems of the CCR flows) are all simple. These C^{∗}-algebras are most like continuous versions of the Cuntz algebras O_{n}, n = 2, 3, …, ∞ (see [3], [10]).

We still do not know if C^{∗}(E) is simple in the cases where U_{E} = ∅. However, we do have the following information in general. Consider the one-parameter unitary group W defined on L^{2}(E) by

W_{t}ξ(x) = e^{itx}ξ(x).

The generator N of W,

W_{t} = e^{itN}

replaces the number operator on ordinary Fock space. N has Lebesgue spectrum distributed throughout [0, ∞) with infinite multiplicity. The associated one parameter group of automorphisms leaves C^{∗}(E) invariant,

W_{t}C^{∗}(E)W_{t}^{∗} = C^{∗}(E), t ∈ ℝ

and thus induces a natural one parameter group of automorphisms γ = {γ_{t} : t ∈ ℝ} of C^{∗}(E). γ is called the gauge group. The best general result in this direction is the following, from which Theorem G is easily deduced (see [3]).

Theorem G1. There are no closed proper ideals J in C^{∗}(E) which are gauge invariant in the sense that γ_{t}(J) = J, for every t ∈ ℝ.

Theorem F tells us that in order to specify an e_{0}-semigroup whose product system is isomorphic to E, it is enough to specify a state of C^{∗}(E). However, there remains a significant question: how does one know when a representation gives rise to an E_{0}- semigroup rather than, say, merely an e_{0}-semigroup? In order to discuss this, let us say that a representation π of C^{∗}(E) is essential if it gives rise to an E_{0}-semigroup, and singular if it gives rise to an e_{0}-semigroup satisfying 7.2. Similarly, a state of C^{∗}(E) (i.e., a nonzero positive linear functional on C^{∗}(E) with no condition on its norm) is called essential or singular according as the representation it defines via the GNS construction has the corresponding property.

The state space P of C^{∗}(E) is a norm-closed cone, and it is known that this cone decomposes into a direct of order ideals

P = E ⊕ S

where E (resp. S) denotes the set of essential (resp. singular) states [4]. A detailed description of this decomposition and the singular summand S is given in [4]. Here, we want to concentrate on the description of the essential summand. In view of the precdding discussion, the assertion that every product system E is associated with an E_{0}-semigroup becomes the assertion that C^{∗}(E) has (nonzero) essential states...i.e., that E ≠ ∅. Our main result along these lines is the following result from [5] which will be discussed further in the following section.

Theorem H. For every product system E, there is an essential state of C^{∗}(E) whose E_{0}-semigroup α is ergodic in the sense that

{A ∈ B(H_{α}) : α_{t}(A) = A, ∀t ≥ 0 } = ℂ1.

Because of Theorem F we have E_{α} ≅ E. This result is analogous to the fact that a locally compact group has irreducible unitary representations. Indeed, the relation that exists between a representation π : C^{∗}(E) → B(H) and its associated e_{0}-semigroup α = {α_{t} : t ≥ 0 } acting on B(H) is expressed in (7.4) and (7.5). From the nature of this relation it follows that the fixed algebra of α,

{A ∈ B(H) : α_{t}(A) = A ∀t ≥ 0 }

is precisely the commutant of π(C^{∗}(E). Thus, the proof of Theorem G amounts to showing that C^{∗}(E) has nonzero essential pure states. Such states will be discussed in the following section.

There are other consequences one can obtain along similar lines. For example, one knows that the C^{∗}-algebraC^{∗}(E) is not GCR, and hence it has representations which generate factors of type II_{∞} or III. There are even essential states with these properties, and hence we may conclude that every E_{0}-semigroup is cocycle conjugate to an E_{0}-semigroup whose fixed agebra is a factor of type II or III.

8. States in the regular representation.

C^{∗}(E) is defined as a C^{∗}-algebra of operators on the Hilbert space L^{2}(E), and this gives rise to a representation λ : E → B(L^{2}(E)). For v ∈ E(t), λ(v) is defined as

λ(v)ξ(x) = \{ \begin{matrix}v · ξ(x − t), & if x > t \\ 0, & if 0 < t < x. \end{matrix}

Notice that for f ∈ L^{1}(E) we have

l_{f} = ∫_{0}^{∞} λ(f(x)) dx.

λ is called the (left) regular representation. The associated e_{0}-semigroup α is singular, since α_{t}(1) is the projection onto to the subspace {ξ ∈ L^{2}(E) : ξ(x) = 0, for 0 < x ≤ t } and these subspaces decrease to 0 as t → ∞. Actually, we will be more concerned with the e_{0}-semigroup generated by the right regular anti representation ρ : E → B(L^{2}(E)), where for v ∈ E(t), ρ(v) is defined as the operator

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

Notice that ρ reverses multiplication in the sense that ρ(uv) = ρ(v)ρ(u), for v, v ∈ E. Nevertheless, we can use ρ to define a second e_{0}-semigroup β = {β_{t} : t ≥ 0 } by way of

β_{t}(A) = ∑_{n=1}^{∞} ρ(e_{n}(t))Aρ(e_{n}(t))^{∗}, A ∈ B(L^{2}(E))

for t > 0, {e_{1}(t), e_{2}(t), … } being any orthonormal basis for E(t), and where β_{0} is defined as the identity endomorphism. It is true (and nontrivial) that β is an e_{0}-semigroup [3],[5].

The generator of β is defined as the limit in the strong operator topology

δ(A) = lim_{t→0+} \frac{1}{t}(A − β_{t}(A)),

A ranging over the set D(δ) of all operators for which the indicated limit exists. It is convenient here to use a different sign in the definition of δ than that of section 2. D(δ) is a unital ∗-subalgebra of B(L^{2}(E)) and δ is an unbounded self adjoint derivation from D(δ) into B(L^{2}(E)). We will first give an alternate description of C^{∗}(E) in terms of δ. This description of C^{∗}(E) is of key importance.

For every t ≥ 0, let P_{t} denote the projection onto the subspace

{ξ ∈ L^{2}(E) : ξ(x) = 0 a.e., for 0 < x ≤ t }.

Notice that α_{t}(1) = β_{t}(1) = P_{t}, and that {P_{t} : t ≥ 0 } is a strongly continuous family of projections that increases from 0 to 1 as t moves from 0 to ∞. An operator A ∈ B(L^{2}(E)) is said to have bounded support if there is a t > 0 such that A = P_{t}AP_{t}, and we write

(8.1)  B_{0} = ⋃_{t>0} P_{t} B(L^{2}(E)) P_{t}

for the ∗-algebra of all operators of bounded support.

Significantly, every operator in B_{0} belongs to the range of δ. Indeed, if A = P_{t}AP_{t}, then the integral defined by the strong limit

I(A) = lim_{T →∞} ∫_{0}^{T} β_{s}(A) ds

exists and obeys

(8.2.1)  ∥I(A)∥ ≤ t,
(8.2.2)  δ(I(A)) = A

(see [5], Theorem 2.2). Indeed, the restriction of I(·) to P_{t}B(L^{2}(E))P_{t} is a normal completely positive linear map for every t > 0.

We will also write

H_{0} = ⋃_{t>0} P_{t}L^{2}(E)

for the linear space of all vectors ξ ∈ L^{2}(E) which have bounded support, and

K_{0} = B_{0} ∩ K

for the ∗-algebra of all compact operators of bounded support.

The following result characterizes C^{∗}(E) in terms of the derivation δ.

Theorem I. The set A of all operators A in the domain of δ satisfying δ(A) ∈ K_{0} is a ∗-algebra whose norm closure is C^{∗}(E).

Recall that C^{∗}(E) is spanned by operators of the form l(f)l(g)^{∗} where f and g are arbitrary integrable sections of E. Notice that if ξ, η are elements of L^{2}(E) which have bounded support, then we may consider ξ and η as elements of L^{1}(E), and operators of the form l(ξ)l(η)^{∗} also span C^{∗}(E). Such an operator belongs to A and the following formula implies that δ(l(ξ)l(η)^{∗} is a rank-one operator in K_{0},

(8.3)  δ(l(ξ)l(η)^{∗}) = ξ ⊗ \overline{η},  ξ, η ∈ H_{0},

ξ ⊗ \overline{η} denoting the operator

ζ ∈ L^{2}(E) ↦< ζ, η > ξ.

The proof of (8.3) can be found in ([5], p. 288).

Theorem I and formula (8.3) open the way to a very explicit description of the state space of C^{∗}(E), which we now describe. Let ω be a linear functional defined on the algebra B_{0} of all operators having bounded support. ω is called a locally normal weight if, for every t > 0, the restriction of ω to P_{t}B(L^{2}(E))P_{t} is a normal positive linear functional. Locally normal weights are generalizations of normal weights (more precisely, of noncommutative Radon measures). Indeed, if

ω : B(L^{2}(E))^{+} → [0, +∞]

is a normal weight satisfying ω(P_{t}) < +∞ for every t > 0, then the restriction of ω to B_{0} is a locally normal weight. We emphasize, however, that not every locally normal weight can be extended to a normal weight of B(L^{2}(E)) (see [5], appendix A for an example). Thus, locally normal weights are more general than normal weights.

Notice that β_{t}(B_{0}) ⊆ B_{0} for every t > 0. Thus we can make the following

Definition 8.4. A locally normal weight ω is called decreasing if for every t ≥ 0 we have

ω(β_{t}(A^{∗}A)) ≤ ω(A^{∗}A), A ∈ B_{0}.

ω is called invariant if equality holds for every t > 0 and every A ∈ B_{0}.

We will call such an ω simply a decreasing weight. Let W denote the cone of all decreasing weights satisfying the growth condition

sup_{t>0} \frac{1}{t}ω(P_{t}) < ∞.

Noting that P_{t} = 1 − β_{t}(1), it is clear that an invariant weight ω satisfies

ω(β_{s}(1) − β_{s+t}(1)) = ω(1 − β_{t}(1))

for all s, t > 0, and from this it follows that there is a constant c ≥ 0 such that

ω(P_{t}) = c · t.

In particular, every invariant weight belongs to W.

We can now describe the state space of C^{∗}(E) (see [5], Theorems 4.15 and 5.7). Every locally normal weight ω defines a linear functional dω on A by way of

dω(A) = ω(δ(A)).

Notice that dω is the “derivative” of ω in the direction of the flow of the e_{0}- semigroup β. We need to know when dω is a positive linear functional which has finite norm. The characterization is as follows.

Theorem J. For every decreasing weight ω ∈ W, dω is a positive linear functional on A having norm

∥dω∥ = sup_{t>0} \frac{1}{t}ω(P_{t}).

ω → dω is an affine order isomorphism of the cone W onto the state space of C^{∗}(E), which maps the subcone of invariant weights onto the cone of essential states.

With Theorem J in hand, it is easy to show that C^{∗}(E) must have essential states. A straightforward construction allows one to write down an invariant weight ω on B_{0} which is normalized so that

ω(P_{t}) = t, ∀t > 0

(see Theorem 5.9 of [5]). It follows from Theorem J that dω is an essential state of C^{∗}(E) satisfying ∥dω∥ = 1.

Corollary. For every abstract product system E, there is an E_{0}-semigroup α for which E_{α} is isomorphic to E.

As we have pointed out previously, with a little care, one can arrange that α is ergodic in the sense that

α_{t}(A) = A, ∀t ≥ 0 ⟹ A = scalar.

The details can be found in [5].

There are numerous interesting unsolved problems concerning the spectral C^{∗}- algebras C^{∗}(E). For example, if E_{n} is the standard product system of dimension n, n = 1, 2, …, ℵ_{0}, then C^{∗}(E) is known t be a “continuous time” analogue of the Cuntz algebra O_{n+1} (see [10]). However, we do not yet know if these C^{∗}-algebras C^{∗}(E_{n}) are mutually non-isomorphic for different values of n. Cerainly this is the case for the O_{n}. In fact, Cuntz showed that O_{m} is not isomorphic to O_{n} essentially by calculating the K-theory of these C^{∗}-algebras [15],[16]. But while the K-theory of the C^{∗}-algebras C^{∗}(E_{n}) has not been computed, there is some evidence that K-theory may not be capable of distinguishing between them.

Finally, we want to point out the remarkable fact that, like O_{∞}, every spectral C^{∗}-algebra C^{∗}(E) has an unbounded trace τ. This is easily seen using the description of C^{∗}(E) given in Theorem I. Let A_{1} denote the set of all operators A in the domain of δ such that δ(A) is a trace class operator of bounded support. A_{1} is a self-adjoint ideal in A which is clearly norm-dense in C^{∗}(E). We can define a linear functional τ on A_{1} by way of

τ(A) = trace(δ(A)).

Notice that τ(AB) = τ(BA), since

trace(δ(AB)) = trace(Aδ(B)) + (δ(A)B) = trace(δ(B)A) + trace(Bδ(A))
= trace(δ(BA)).

It is possible to show that τ is not a positive trace in that there exist operators A ∈ A_{1} satisfying τ(A^{∗}A) < 0. Such traces are uncommon in operator theory, but notice that the Wodzicki residue provides another example of an unbounded non-positive trace on an algebra of pseudo-differential operators [14]. As yet, the role of this trace in the theory of E_{0}-semigroups remains mysterious.

References

1. Araki, H. and Woods, E. J., Complete Boolean algebras of type I factors, Publ. RIMS (Kyoto University) 2, ser. A, no. 2 (1966), 157–242.

2. Arveson, W., Continuous analogues of Fock space, Memoirs Amer. Math. Soc. 80 no. 3 (1989).

3. , Continuous analogues of Fock space II: the spectral C^{∗}-algebra, J. Funct. Anal. 90 (1990), 138–205.

4. , Continuous analogues of Fock space III: singular states, J. Oper. Th. 22 (1989), 165–205.

5. , Continuous analogues of Fock space IV: essential states, Acta Math. 164 (1990), 265–300.

6. , An addition formula for the index of semigroups of endormorphisms of B(H), Pac. J. Math. 137 (1989), 19–36.

7. , Quantizing the Fredholm index, Operator Theory: Proceedings of the 1988 GPOTS-Wabash conference (Conway, J. B. and Morrel, B. B., ed.), Pitman research notes in mathematics series, Longman, 1990.

8. , The spectral C^{∗}-algebra of an E_{0}-semigoup, Operator Theory Operator Algebras and applications, Proc. Symp. Pure Math. (Arveson, W. and Douglas, R. G., ed.), vol. 51, part I, 1990, pp. 1–15.

9. , Decomposable E_{0}-semigroups, (in preparation).

10. , C^{∗}-algebras associated with sets of semigroups of isometries, Int. J. Math. 2, no. 3 (1991).

11. Arveson, W. and Kishimoto, A., A note on extensions of semigroups of ∗-endomorphisms, Proc. A. M. S. 116, no 3 (1992), 769–774.

12. Bratteli, O. and Robinson, D. W., Operator algebras and quantum statistical mechanics I, II, Springer-Verlag, 1989.

13. Connes, A., Une classification des facteurs de type III, Ann. Scient. Ecole Norm. Sup. 6 , fasc. 2 ser. 4e (1973), 133–253.

14. , Non Commutative Geometry, Academic Press (to appear).

15. Cuntz, J., Simple C^{∗}-algebras generated by isometries, Comm. Math. Phys. 57 (1977), 173– 185.

16. , K-theory for certain C^{∗}-algebras, Ann. Math. 113 (1981), 181–197.

17. Pedersen, G. K., C^{∗}-algebras and their automorphism groups, Academic Press, 1979.

18. Powers, R. T., An index theory for semigroups of endomorphisms of B(H) and type II factors, Can. J. Math. 40 (1988), 86–114.

19. A non-spatial continuous semigroup os ∗-endomorphisms of B(H), Publ. RIMS (Kyoto University) 23 (1987), 1053–1069.

20. On the structure of continuous spatial semigroups of ∗-endomorphisms of B(H), Int. J. Math. 2, no 3 (1991), 323–360.

21. , New examples of continuous spatial semigroups of endomorphisms of B(H), (preprint 1994).

22. Powers, R. T. and Robinson, D., An index for continuous semigroups of ∗-endomorphisms of B(H), J. Funct. Anal. 84 (1989), 85–96.

23. Powers, R. T. and Price, G, Continuous spatial semigroups of ∗-endomorphisms of B(H), Trans. A. M. S. (to appear).