DYNAMICAL INVARIANTS FOR NONCOMMUTATIVE FLOWS

William Arveson

Department of Mathematics University of California Berkeley CA 94720, USA

18 November 1995

Abstract. We show that semigroups of endomorphisms of B(H) can often be associated with a dynamical principle; that is, an infinitesimal structure that completely determines the dynamics. These dynamical invariants are similar to the second order differential equations of classical mechanics, in that one can locate a space of momentum operators, a “Riemannian metric”, and a potential. In the simplest cases these structures occur in n × n matrix algebras, and can be classified in terms of noncommutative geometric invariants. As a consequence, we obtain new information relating to the classification of E_{0}-semigroups acting on type I_{∞} factors.

Contents

Introduction 1. Completely positive semigroups

2. Differential operators on matrix algebras

3. Semielliptic operators

4. Momentum

5. Modular Cohomology

6. Exactness

7. Classification of Semielliptic operators

8. Applications References

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

Markov semigroups. This research was supported by NSF grants DMS92-43893 and DMS95-00291

Typeset by AMS-TEX

Introduction. The dynamical groups of quantum theory tend to be one-parameter groups of automorphisms of C^{∗}-algebras generated by bounded functions of certain observables. Frequently, there is a distinguished state on the algebra that is invariant under the action of the automorphism group and which represents the vacuum. Using a standard GNS construction, one can arrange that the operator algebra acts on a Hilbert space, that the automorphism group is implemented by a oneparameter group of unitary operators on this representation space, and that there is a unit vector in the Hilbert space, fixed under the action of the unitary group, which gives rise to the vacuum state.

We have settled on the following general context for confronting dynamical systems of this type.

Definition A. A history is a triple (M_{−}, U, v) consisting of a one-paramter group of unitary operators U = {U_{t} : t ∈ ℝ} acting on a separable Hilbert space H, a unit vector v ∈ H satisfying

U_{t}v = v, t ∈ ℝ,

and a von Neumann algebra M_{−} ⊆ B(H) such that

γ_{t}(M_{−}) ⊆ M_{−}, t ≤ 0,

γ = {γ_{t} : t ∈ ℝ} denoting the automorphism group of B(H) associated with U.

The group γ acts on B(H) by

(0.1)  γ_{t}(a) = U_{t}aU_{t}^{∗},  t ∈ ℝ.

There are two kinds of degeneracies that we have found it necessary to rule out. Suppose first that M_{−} is an abelian von Neumann algebra. Then so is the larger von Neumann algebra

M = (⋃_{t≥0} γ_{t}(M_{−}))′′

and M is invariant under the full dynamical group {γ_{t} : t ∈ ℝ}. After introducing appropriate coordinates we may realize M as the algebra of all bounded random variables on a standard probability space (Ω, P), in such a way that the group action γ is represented by a one-parameter group of measure-preserving transformations of (Ω, P). M_{−} becomes a subalgebra of L^{∞}(Ω, P) that is invariant under negative time translations.

Now assume further that there is a von Neumann algebra M_{0} ⊆ M_{−} such that M_{−} is generated by the negative time translations of M_{0}:

(0.2)  M_{−} = (⋃_{t≤0} γ_{t}(M_{0}))′′.

In this case we can find a single random variable

X_{0} : Ω → ℝ

with the property that M_{0} consists of all bounded measurable functions of X_{0} (we may choose X_{0} to be bounded if we wish). We can define a stationary stochastic process {X_{t} : t ∈ ℝ} in the obvious way

X_{t} = γ_{t}(X_{0}), t ∈ ℝ,

and thus M_{−} appears as the “past” of the process {X_{t} : t ∈ ℝ}. Thus this case falls completely into the domain of stationary stochastic processes.

In general it may be impossible to find a “sharp time” subalgebra M_{0} which satisfies (0.2). Nevertheless, that situation too is familiar in modern probability theory. For example, that would be the the case when M_{−} is the “past” subalgebra associated with a stationary random distribution such as white noise.

For our purposes this probabilistic case is a degenerate one, and we rule it out by imposing the following requirement that forces us to the opposite extreme:

Definition B. A History (M_{−}, U, v) is called primary if M_{−} is a factor.

The second type of degeneracy represents a kind of determinism. Let (M_{−}, U, v) be a history. Then M_{−} is invariant under γ_{t} for every t ≤ 0, and if there is a single t_{0} < 0 for which γ_{t_{0}}(M_{−}) = M_{−}, then γ_{t}(M_{−}) = M_{−} for every negative t and in fact M_{−} is invariant under the full dynamical group {γ_{t} : t ∈ ℝ}. One may interpret this as a deterministic situation in which the future is a function of the past (more precisely, every observable is a Borel function of an observable belonging to M_{−}). From our point of view, however, this simply represents a bad choice of the subalgebra M_{−}.

These considerations have led us to the following program for understanding the nature of noncommutative dynamical systems. One should seek to classify primary histories (M_{−}, U, v) which satisfy the condition

γ_{t}(M_{−}) ⫋ M_{−}, for every t < 0.

There are two significant equivalence relations for histories. The first is an obvious notion of conjugacy: (M_{−}, U, v) and (M̃_{−}, Ũ, ṽ) are said to be conjugate if there is a unitary operator W : H → H̃ such that

(i)  W v = ṽ
(ii)  W U_{t} = Ũ_{t}W,  t ∈ R
(iii)  W M_{−}W^{−1} = M̃_{−}.

The second equivalence relation is cocycle conjugacy, and will be taken up elsewhere.

There are several ways of finding histories among other objects that arise naturally in quantum physics. For example, suppose that we start quite simply with a pair (U, v) consisting of a one-parameter unitary group U acting on a Hilbert space H and a unit vector v ∈ H such that U_{t}v = v for every t ∈ ℝ. Assume further that K is another Hilbert space and we are given a representation W_{0} of the canonical commutation relations over K. Thus

f ∈ K ↦ W_{0}(f) ∈ B(H)

is a strongly continuous mapping from K to the unitary operators on H for which

W_{0}(f)W_{0}(g) = e^{iω(f,g)}W_{0}(f + g), f, g ∈ K,

ω(f, g) denoting the imaginary part of the inner product ⟨f, g⟩ in K. Then for every t ∈ ℝ we can define a representation W_{t} : K → B(H) of the CCRs by W_{t}(f) = U_{t}W_{0}(f)U_{t}^{∗}. One may think of the family {W_{t} : t ∈ ℝ} as a noncommutative quantum process. In this case we have a natural candidate for M_{−}:

M_{−} = (⋃_{t≤0} W_{t}(K))′′,

and thus we have a history (M_{−}, U, v).

In this example there is a “sharp time” von Neumann algebra associated with time zero

M_{0} = W_{0}(K)′′

with the property that M_{−} is generated by M_{0} and its negative time images under the group of automorphisms associated with U. In general, one cannot expect to have sharp time algebras as this example does. Nevertheless, in very general circumstances (e.g., in any case where one has a quantum system obeying the Haag-Kastler axioms [16, pp. 106–107]) one can find significant examples of histories (M_{−}, U, v).

This paper deals with the classification of histories (M_{−}, U, v) of the simplest type, namely

(0.3.1) M_{−} is a factor of type I, and

(0.3.2) γ_{t}(M_{−}) ⫋ M_{−}, for every t < 0.

This leads to the problem of classifying pairs of E_{0}-semigroups in the following way.

Let M_{+} be the commutant of M_{−}. Then we obtain a pair of E_{0}-semigroups α^{+}, α^{−} acting respectively on M_{+} and M_{−} by

α_{t}^{+}(a) = U_{t}aU_{t}^{∗},  t ≥ 0, a ∈ M_{+}
α_{t}^{−}(b) = U_{t}^{∗}bU_{t},  t ≥ 0, b ∈ M_{−}.

Moreover, the vector state of B(H) defined by v, ω(a) = ⟨av, v⟩, can be restricted to these two subalgebras to give normal states ρ^{+}, ρ^{−} which are invariant under the actions of the respective E_{0}-semigroups. Thus the classification problem for type I histories reduces to the problem of classifying pairs of E_{0}-semigroups which have distinguished normal invariant states. The theory of E_{0}-semigroups was initiated by Powers in [24], and has undergone considerable development during the past several years, see [1], [3], [4], [5], [25], [26], [27] and references cited in these papers.

Notice too that while ω(a) = ⟨av, v⟩ is a pure state of B(H) its restrictions to the subfactors M_{+} and M_{−} are not necessarily pure states on these algebras. Thus we are led to the problem of classifying pairs (α, ρ) where α = {α_{t} : t ≥ 0} is an E_{0}-semigroup acting on a a type I_{∞} factor (which we may take as B(H_{α}) for some separable Hilbert space H_{α}) and ρ is a normal state on B(H_{α}) satisfying

ρ(α_{t}(a)) = ρ(a), a ∈ B(H_{α}), t ≥ 0.

Two such pairs (α, ρ) and (α̃, ρ̃) are said to be conjugate if there exists a ∗- isomorphism θ : B(H_{α}) → B(Hα̃) such that

α̃_{t} ∘ θ = θ ∘ α_{t},  t ≥ 0,
ρ̃ ∘ θ = ρ.

Our main results below apply to the case where where ρ is not a pure state and is weakly continuous. Equivalently, if we realize ρ in the form

ρ(a) = trace(Ωa)

where Ω is a positive trace-class operator on H_{α}, then we are assuming that

(0.4)  2 ≤ rank Ω < ∞.

In the extreme case where Ω is of rank 1, ρ is a pure state of B(H_{α}) and in this case all of our invariants become trivial. However, we point out that Powers has recently worked out a standard form for E_{0}-semigroups [26] and any E_{0}-semigroup in standard form admits an invariant vector state. Thus the case where Ω is rank one should not be considered mysterious. The other extreme case (Ω is of infinite rank) appears to be amenable to the techniques developed below, but there are technical difficulties associated with the infinite rank case that require special attention. We have elected to postpone discussion of the infinite rank case to a subsequent paper.

In broad terms, our results show that there is an infinitesimal “dynamical principle” that governs the behavior of such a pair (α, ρ). This invariant is analogous to the Hamiltonian structure (on the cotangent bundle of a Riemannian manifold) that governs the behavior of a constrained classical mechanical system with a finite number of degrees of freedom. Indeed, the work carried out in the following sections amounts to a classification of Riemannian type structures in finite dimensional matrix algebras.

The geometric nature of these invariants shows that the “dynamical principles” that govern the behavior of E_{0}-semigroups are closely akin to second order differential equations. As in classical mechanics, we are able to identify two fundamental aspects of the dynamics: a metric term corresponding to momentum and kinetic energy, and a potential term corresponding to the driving force. Moreover, one is free to specify the momentum space, the metric, and the potential operator arbitrarily. Regardless of how this is done one obtains an E_{0}-semigroup pair (α, ρ) (see remark 8.8 for more detail). In particular, this gives an entirely new way of constructing E_{0}-semigroups.

On the other hand, not all pairs (α, ρ) (satisfying the finiteness condition (0.4)) arise in this way. But we are able to give a useful characterization of those that do. Broadly speaking, a pair (α, ρ) arises in this way iff it satisfies two conditions. First, α must obey a certain minimality condition (which can always be arranged by replacing α with a compression to a suitable hereditary subalgebra of B(H_{α})). More significantly, (α, ρ) must also be exact in an appropriate sense. Exactness is equivalent to several important properties, and it is closely analogous to the hypothesis of classical mechanics in which one supposes that the force (a vector field on configuration space) should be the gradient of a potential.

1. Markov semigroups. The purpose of this section is to make some observations about the relationship that exists between E_{0}-semigroups and semigroups of completely positive maps.

Let ϕ = {ϕ_{t} : t ≥ 0} be a semigroup of normal completely positive linear maps of B(H) such that ϕ_{t}(1) = 1, and which is continuous in the sense that for every a ∈ B(H), ξ, η ∈ H, ⟨ϕ_{t}(a)ξ, η⟩ is a continuous function of t. We will refer to such a semigroup simply as a completely positive semigroup.

Definition 1.1. A Markov semigroup is a pair (ϕ, ρ) consisting of a completely positive semigroup ϕ acting on B(H) and a faithful normal state ρ of B(H) which is invariant under ϕ in that ρ ∘ ϕ_{t} = ρ for every t.

Remark. We emphasize that the normal state should be faithful: ρ(a^{∗}a) = 0 ⟹ a = 0, for every a ∈ B(H).

Two Markov semigroups (ϕ, ρ), (ϕ′, ρ′) are said to be conjugate if there is a ∗-isomorphism θ : B(H) → B(H′) such that

(1.2.1)  θ ∘ ϕ_{t} = ϕ′_{t} ∘ θ,  t ≥ 0,
(1.2.2)  ρ′ ∘ θ = ρ.

In this paper, we will be primarily concerned with Markov semigroups acting on finite dimensional spaces; that is, Markov semigroups which act on n × n matrix algebras, 2 ≤ n < ∞.

The connection between Markov semigroups and E_{0}-semigroups is based on the following observations. By an E_{0}-semigroup we mean a semigroup α = {α_{t} : t ≥ 0} of normal ∗-endomorphisms of the algebra B(H) of all bounded operators on a separable Hilbert space H, such that α_{t}(1) = 1 and which is continuous in the sense described above.

Lemma 1.3. Let α be an E_{0}-semigroup acting on B(H) and let ω be a normal state of B(H) which is invariant under the action of α_{t} for every t ≥ 0. Let p_{0} be the support projection of ω. Then α_{t}(p_{0}) ≥ p_{0} for every t ≥ 0.

Remark. The support projection of ω is the smallest projection p with the property that ω(1 − p) = 0. One has ω(a) = ω(p_{0}ap_{0}) for every a ∈ B(H), and ω(a^{∗}a) = 0 iff ap_{0} = 0.

proof of 1.3. Let t ≥ 0, and consider the projection α_{t}(p_{0}). Because of the invariance of ω we have

ω(1 − α_{t}(p_{0})) = ω ∘ α_{t}(1 − p_{0}) = ω(1 − p_{0}) = 0,

and hence α_{t}(p_{0}) ≥ p_{0} □

It follows immediately that the family of projections p_{t} = α_{t}(p_{0}) is increasing on the interval 0 ≤ t < ∞. Let us write M for the von Neumann algebra B(H). Then we can define a family ϕ = {ϕ_{t} : t ≥ 0} of completely positive maps on the hereditary subalgebra M_{0} = p_{0}Mp_{0} as follows:

ϕ_{t}(a) = p_{0}α_{t}(a)p_{0}, a ∈ M_{0}.

We have ϕ_{t}(p_{0}) = p_{0}, and the semigroup property follows from the observation that if s, t ≥ 0 then

ϕ_{s}(ϕ_{t}(a)) = p_{0}α_{s}(p_{0}α_{t}(a)p_{0})p_{0} = p_{0}p_{s}α_{s+t}(a)p_{s}p_{0} = ϕ_{s+t}(a),

because p_{0}p_{s} = p_{s}p_{0} = p_{0}. Thus, if we let ρ be the restriction of ω to M_{0} and then identify M_{0} with B(p_{0}H), we find that we have a Markov semigroup (ϕ, ρ). If ω is weakly continuous (equivalently, p_{0} is finite dimensional) then we may consider that (ϕ, ρ) acts on an n × n matrix algebra.

Notice that if ω happens to be a pure (vector) state, then p_{0}H is one-dimensional and ϕ is the trivial semigroup. Thus all information has been lost in the passage from (α, ω) to (ϕ, ρ). On the other hand, in all other cases the Markov semigroup will contain significant information about (α, ω).

Indeed, it is often the case that (ϕ, ρ) completely determines (α, ω). In order to discuss this issue briefly, let H_{0} = p_{0}H, and consider the subspace H_{+} ⊆ H defined by

H_{+} = [α_{t_{1}}(a_{1})α_{t_{2}}(a_{2}) … α_{t_{n}}(a_{n})ξ : a_{i} ∈ M_{0}, ξ ∈ H_{0}, n = 1, 2, …].

There is a corresponding von Neumann subalgebra M_{+} of B(H), namely the weakly closed algebra generated by the family of operators {α_{t}(a) : a ∈ M_{0}, t ≥ 0}. It can be shown that M_{+} is a hereditary subalgebra of B(H), and in fact

M_{+} = p_{+}B(H)p_{+},

p_{+} denoting the projection of H onto H_{+}. The original pair (α, ω) is called minimal if H_{+} = H, or equivalently M_{+} = B(H).

If (α, ω) is minimal, then it is determined up to conjugacy by its associated Markov semigroup (ϕ, ρ) (this follows from work of Bhat described in the following paragraph). But even when (α, ρ) is not minimal, one may compress α to the invariant subalgebra M_{+} = p_{+}B(H)p_{+} to obtain a minimal pair (α′, ω′). Thus one may always assume that minimality is satisfied provided one is willing to compress α to an invariant corner of B(H).

Moreover, a recent dilation theorem of B. V. R. Bhat [6], building on and clarifying earlier partial results ([17], [18], [19], [11], [14], [21]), implies that every Markov semigroup (α, ρ) arises in this way from a minimal pair (α, ω), where α is an E_{0}- semigroup and ρ is a normal α-invariant state. We will discuss Bhat’s theorem, minimality criteria, and properties of the “n-point functions”

ω(α_{t_{1}}(a_{1})α_{t_{2}}(a_{2}) … α_{t_{n}}(a_{n}))

defined for a_{1}, …, a_{n} ∈ M_{0}, t_{i} ≥ 0, n = 1, 2, … in a subsequent paper.

2. Differential operators on matrix algebras.

There is a natural notion of the order of a differential operator which is easily adapted to the context of linear operators on arbitrary commutative algebras. Indeed, if A is a complex commutative algebra and L : A → A is a linear operator, one may define a sequence of multilinear mappings Δ^{n}L : A^{n+1} → A as follows: Δ^{0}L = L and

Δ^{n+1}L(f_{1}, …, f_{n}, f_{n+1}; a) = Δ^{n}L(f_{1}, …, f_{n}; f_{n+1}a) − f_{n+1}Δ^{n}L(f_{1}, …, f_{n−1}; a).

One finds that Δ^{n}L(f_{1}, …, f_{n}; a) is a symmetric function of its first n variables f_{1}, …, f_{n}. L is said to be a differential operator of order n if Δ^{n+1}L = 0 and Δ^{n}L ≠ 0.

The purpose of this section is to discuss the extent to which the notion of order is appropriate for operators on noncommutative algebras, and to introduce a suitable definition of symbol of a differential operator. For definiteness, we will consider the case where A is the algebra of all n × n matrices over ℂ, since it is this case that is relevant for our purposes. However, the reader will note that we make no essential use of structures specific to M_{n}(ℂ). Throughout, L(A) will denote the algebra of all linear operators L : A → A.

Definition. L is said to be a first order differential operator if for every a, x, y ∈ A we have

(2.1)  L(xay) − xL(ay) − L(xa)y + xL(a)y = 0.

Notice that the trilinear form appearing on the left side of (2.1) is a noncommutative version of the trilinear form Δ^{2}L(x, y; a) discussed above. Moreover, since A has a unit (2.1) is equivalent to the somewhat simpler condition

L(xy) − xL(y) − L(x)y + xL(1)y = 0

for all x, y ∈ A (this will be discussed more fully later in the section).

A simple computation shows that any operator of the form

(2.2)  L(x) = D(x) + ax

where a is a fixed element of A and D is a derivation must be a first order differential operator. The fact that the multiplier a appears on the left in (2.2) is inessential, since any operator of the form L(x) = D(x) + ax + xb can be put into the form (2.2) by replacing D with the derivation D′(x) = D(x) + xb − bx. Noting that the first order differential operators L satisfying L(1) = 0 are derivations, it follows that the first order differential operators are exactly those of the form (2.2).

In contrast to the commutative case, operators having the form L(x) = ax or L′(x) = xa should not be regarded as “order zero” differential operators, since the difference of two such operators L(x) − L′(x) = ax − xa is in this case a nontrivial derivation which must be assigned order 1. Thus there is no meaningful definition of “order zero” for operators on noncommutative algebras.

Similarly, there is no viable concept of “order n” for n > 2. For in the case where A = M_{n}(ℂ) is a matrix algebra, every linear operator L on A is a finite sum of double multipliers x ↦ axb, a, b being fixed elements of A, and thus by the preceding paragraph L has the form

L = A_{1}B_{1} + A_{2}B_{2} + ⋯ + A_{r}B_{r},

where A_{k} and B_{k} are first order differential operators. We conclude that every operator on A is of “order” at most 2.

Despite the somewhat negative tone of these remarks, we will find Definition (2.1) to be quite useful. In order to discuss this issue, we recall the differential algebra Ω^{∗}(A) associated with A. This was introduced in [2], and has become a basic constituent of Connes’ noncommutative differential calculus [9]. Actually, we only require the two modules Ω^{1}(A) and Ω^{2}(A), which can be defined for unital ∗-algebras as follows. Consider the tensor product A ⊗ A as an involutive bimodule over A, with

a(x ⊗ y)b = ax ⊗ yb,
(x ⊗ y)^{∗} = y^{∗} ⊗ x^{∗}.

The map d : A → A ⊗ A defined by dx = 1 ⊗ x − x ⊗ 1 is a derivation for which (dx)^{∗} = −d(x^{∗}), and it is a universal derivation of A in the sense that if E is any bimodule and D : A → E is a derivation, then there is a unique homomorphism of bimodules θ : Ω^{1}(A) → E such that θ ∘ d = D. Every element of Ω^{1}(A) is a finite sum of the form

ω = ∑_{k=1}^{r} a_{k} dx_{k}.

Finally, Ω^{1}(A) is the kernel of the multiplication map μ : A ⊗ A → A defined by μ(a ⊗ b) = ab, and thus we have an exact sequence of involutive bimodules

(2.3)  0 ⟶ Ω^{1}(A) ⟶ A ⊗ A ⟶_{μ} A ⟶ 0.

For brevity, we will use the notation Ω^{1}, Ω^{2} rather than Ω^{1}(A), Ω^{2}(A) whenever it does not lead to confusion. Ω^{2} is defined by

Ω^{2} = Ω^{1} ⊗_{A} Ω^{1},

and every element of Ω^{2} is a finite sum of the form

ω = ∑_{k=1}^{r} a_{k} dx_{k} dy_{k}.

Alternately, Ω^{2} can be viewed as the submodule of the bimodule A^{⊗3} = A⊗A⊗A that is linearly spanned by elements of the form

adxdy = a(1 ⊗ x − x ⊗ 1)(1 ⊗ y − y ⊗ 1)
= a ⊗ x ⊗ y − ax ⊗ 1 ⊗ y + ax ⊗ y ⊗ 1 − a ⊗ xy ⊗ 1.

There are two multiplication maps defined on A^{⊗3} which together give rise to a homomorphism of bimodules μ : A^{⊗3} → A^{⊗2} ⊕ A^{⊗2}:

μ(a ⊗ b ⊗ c) = (ab ⊗ c, a ⊗ bc),

and we have an exact sequence of modules

(2.4)  0 ⟶ Ω^{2} ⟶ A^{⊗3} ⟶_{μ} A^{⊗2} ⊕ A^{⊗2} ⟶ 0.

The involutions defined in A^{⊗3} and A^{⊗2} ⊕ A^{⊗2} by

(a ⊗ b ⊗ c)^{∗} = c^{∗} ⊗ b^{∗} ⊗ a^{∗}
(ξ, η)^{∗} = (η^{∗}, ξ^{∗})

make (2.4) into an exact sequence of involutive bimodules.

We will make frequent use of the following observation.

Lemma 2.5. Let L ∈ L(A). There is a unique homomorphism of bimodules θ_{L} ∈ hom(Ω^{2}, A) such that

θ_{L}(dx dy) = L(xy) − xL(y) − L(x)y + xL(1)y.

proof. The uniqueness of θ_{L} is clear from the fact that Ω^{2} is spanned by elements of the form a dx dy, and

θ_{L}(a dx dy) = a(L(xy) − xL(y) − L(x)y + xL(1)y).

For existence, we can define θ ∈ hom(A^{⊗3}, A) by θ(a ⊗ b ⊗ c) = −aL(x)b, and one finds that the restriction of θ to elements of the form dx dy is as required □

A straightforward computation shows that, more generally,

θ_{L}(a dx b dy c) = aL(xby)c − axL(by)c − aL(xb)yc + axL(b)yc.

Note too that L is a first order differential operator if, and only if, θ_{L} = 0.

Now it follows from the definition of Ω^{1} that for any derivation D of A there is a unique θ_{D} ∈ hom(Ω^{1}, A) such that

θ_{D}(dx) = D(x), x ∈ A.

Moreover, the map D ↦ θ_{D} is a linear isomorphism of the space of derivations of A onto hom(Ω^{1}, A). There is a somewhat similar characterization of hom(Ω^{2}, A), valid for any unital algebra A for which the Hochschild cohomology space H^{2}(A, A) is trivial:

Lemma 2.6. Let θ ∈ hom(Ω^{2}, A). There is a linear operator L ∈ L(A) satisfying L(1) = 0 and θ = θ_{L}. L is unique up to a perturbation of the form L′ = L + D where D is a derivation of A.

proof. Fix θ, and consider the bilinear map T : A × A → A defined by T (x, y) = θ(dx dy). The Hochschild coboundary of T

bT (x, y, z) = xT (y, z) − T (xy, z) + T (x, yz) − T (x, y)z
= θ(x dy dz − d(xy) dz + dx d(yz) − dx dy z)

vanishes because

d(xy) dz = x dy dz + dx y dz,  and
dx d(yz) = dx y dz + dx dy z.

Since H^{2}(A, A) = 0 when A is a matrix algebra, there exists a linear operator L ∈ L(A) for which

θ(dx dy) = bL(x, y) = xL(y) − L(xy) + L(x)y.

Setting L_{0}(x) = L(x) − xL(1) we find that L_{0}(1) = 0 and θ = θ_{−L_{0}}.

If L_{1} and L_{2} are two linear operators satisfying L_{1}(1) = L_{2}(1) = 0 and θ = θ_{L_{1}} = θ_{L_{2}}, then θ_{L_{1}−L_{2}} = 0. Since L_{1} − L_{2} is a first order differential operator that annihilates 1, it must be a derivation □

These remarks about the differential nature of linear operators on A are summarized in the following exact sequence of complex vector spaces, in which L_{0}(A) = {L ∈ L(A) : L(1) = 0}, ι is the identification of hom(Ω^{1}, A) with derivations by way of ιθ(x) = θ(dx) x ∈ A, and θ : L ↦ θ_{L} is the mapping of Lemma 2.5,

0 ⟶ hom(Ω^{1}, A) ⟶_{ι} L_{0}(A) ⟶_{θ} hom(Ω^{2}, A) ⟶ 0.

Throughout the remainder of this paper we will be concerned with pairs (A, ρ) consisting of a finite dimensional C^{∗}-algebra A (usually M_{n}(ℂ)) together with a distinguished faithful state ρ, that is a linear functional satisfying

ρ(1) = 1,  and
ρ(a^{∗}a) > 0,  for every nonzero a ∈ A.

The associated space D(A, ρ) of differential operators is defined by the three requirements:

(2.7.1)  Normalization:  L(1) = 0
(2.7.2)  Divergence zero:  ρ ∘ L = 0
(2.7.3)  Symmetry:  L(x^{∗}) = L(x)^{∗}, x ∈ A.

Conditions (2.7.1) and (2.7.2) are dual to each other, in a sense that will be exploited in the following sections. In general, an operator L satisfying (2.7.3) will be called symmetric. D(A, ρ) is a real vector space of linear operators on A. For every L ∈ D(A, ρ) we define the symbol of L by σ_{L} = ρ ∘ θ_{L}. More explicitly, σ_{L} is the linear functional defined on Ω^{2} by

σ_{L}(a dx dy) = ρ(aL(xy) − axL(y) − aL(x)y).

We collect the following elementary properties of the symbol for later use.

Proposition 2.9. For every L ∈ D(A, ρ), σ_{L} has the following properties:

(i)  σ_{L}(ξ^{∗}) = \overline{σ_{L}(ξ)},  ξ ∈ Ω^{2}
(ii)  σ_{L} = 0 iff L is a derivation.

proof. (i) follows from the formula θ_{L}(ξ^{∗}) = θ_{L}(ξ)^{∗}, which in turn reduces to the property L(x^{∗}) = L(x)^{∗} after evaluating both sides at elements of the form ξ = dx dy.

For (ii), we note that the bilinear form a, b ∈ A ↦ ρ(ab) is nondegenerate because ρ is a faithful state. Thus θ_{L}(dx dy) = 0 for all x, y ∈ A iff ρ(aθ_{L}(dx dy)) = 0 for every a, x, y ∈ A, hence (ii). □

Finally, we note that for any finite set D_{0}, D_{1}, …, D_{r} of symmetric derivations of A satisfying ρ ∘ D_{k} = 0 for 0 ≤ k ≤ r, the operator

L(x) = D_{0}(x) + ∑_{k=1}^{r} D_{k}^{2}(x)

belongs to D(A, ρ), and has symbol

(2.10)  σ_{L}(a dx dy) = 2 ∑_{k=1}^{r} ρ(aD_{k}(x)D_{k}(y)).

More generally, one has

(2.11)  σ_{L}(a dx b dy c) = 2 ∑_{k=1}^{r} ρ(aD_{k}(x)bD_{k}(y)c).

It is operators of this type that will be central to our analysis of Markov semigroups.

3. Semielliptic operators. Throughout this section, A will denote a finite dimensional matrix algebra and (ϕ, ρ) will denote a Markov semigroup acting on A. In this case ϕ is uniformly continuous

lim_{t→0} ∥ϕ_{t} − id∥ = 0.

The infinitesimal generator L exists in all senses, and obeys the three properties (2.7). Hence L belongs to D(A, ρ). There are two characterizations of the generators of completely positive semigroups that are significant for our purposes. The first is due to Lindblad [20] and independently to Gorini et al [15] (also see [10, Theorem 4.2]). The second characterization is due to Evans and Lewis [14], based on work of Evans [12]. These two results can be paraphrased in our context as follows.

Theorem 3.1. An operator L ∈ D(A, ρ) generates a Markov semigroup iff there is a completely positive linear map P : A → A and an operator a ∈ A such that

L(x) = P (x) + ax + xa^{∗}, x ∈ A.

Theorem 3.2. An operator L ∈ D(A, ρ) generates a Markov semigroup iff for every n ≥ 1 and every set of elements a_{1}, b_{1}, …, a_{n}, b_{n} ∈ A, satisfying b_{1}a_{1} + b_{2}a_{2} + ⋯ + b_{n}a_{n} = 0 we have

∑_{i,j=1}^{n} a_{j}^{∗}L(b_{j}^{∗}b_{i})a_{i} ≥ 0.

An operator L satisfying the condition of (3.2) is called conditionally completely positive [13]. While 3.1 tells us exactly which operators are generators of Markov semigroups, the cited decomposition of L into a sum of more familiar operators is unfortunately not unique.

The purpose of this section is to use (3.2) to give a new characterization of generators of Markov semigroups in terms of their symbols σ_{L} : Ω^{2}(A) → ℂ. Recall that the involution in Ω^{1} is defined by

(a dx)^{∗} = −dx^{∗} a^{∗} = −d(x^{∗}a^{∗}) + x^{∗} da^{∗},

while that of Ω^{2} is defined by

(a dx dy)^{∗} = dy^{∗} dx^{∗} a^{∗}.

Thus for any two 1-forms ω_{2}, ω_{2} ∈ Ω^{1} we can form various products in Ω^{2}: ω_{1}ω_{2}, ω_{1}^{∗}ω_{2}, etc.

Theorem 3.3. An operator L ∈ D(A, ρ) generates a Markov semigroup on (A, ρ) iff σ_{L}(ω^{∗}ω) ≤ 0 for every ω ∈ Ω^{1}.

proof. Let L be an arbitrary operator in L(A), and let θ_{L} : Ω^{2} → A be the module homomorphism defined in Lemma 2.5. Choose any sequence of elements a_{1}, b_{1}, …, a_{n}, b_{n} ∈ A satisfying ∑_{k} b_{k}a_{k} = 0 and define an element ω ∈ A ⊗ A by

(3.5)  ω = ∑_{k=1}^{n} b_{k} ⊗ a_{k}.

Notice that ω belongs to the kernel of the multiplication map μ : A ⊗ A → A and hence ω ∈ Ω^{1}. Conversely, every element ω ∈ Ω^{1} can be deomposed into a sum of the form (3.5) which belongs to the kernel of the multiplication map. Now for such a 1-form ω we have

ω^{∗}ω = ∑_{i,j=1}^{n} a_{j}^{∗} ⊗ b_{j}^{∗}b_{i} ⊗ a_{i}.

Let θ ∈ hom(Ω^{2}, A) be defined by θ(a ⊗ x ⊗ b) = −aL(x)b. The proof of Lemma 2.5 shows that θ_{L} is obtained by restricting θ to Ω^{2}, hence

θ_{L}(ω^{∗}ω) = − ∑_{i,j=1}^{n} a_{j}^{∗}L(b_{j}^{∗}b_{i})a_{i}.

This observation, together with Theorem 3.2, shows that an operator L ∈ D(A, ρ) generates a Markov semigroup if, and only if, θ_{L}(ω^{∗}ω) ≤ 0 for every ω ∈ Ω^{1}. It follows that for every generator L and every ω ∈ Ω^{1} we have σ_{L}(ω^{∗}ω) = ρ(θ_{L}(ω^{∗}ω) ≤ 0.

Conversely, if σ_{L}(ω^{∗}ω) ≤ 0 for every ω ∈ Ω^{1} then for every b ∈ A we have

ρ(b^{∗}θ_{L}(ω^{∗}ω)b) = ρ(θ_{L}((ωb)^{∗}ωb)) = σ_{L}((ωb)^{∗}ωb) ≤ 0.

Since ρ is a faithful state and the operator T = θ_{L}(ω^{∗}ω) obeys ρ(b^{∗}T b) ≤ 0 for every b, it follows that T ≤ 0. Hence θ_{L}(ω^{∗}ω) ≤ 0 for every ω ∈ Ω^{1} and we may conclude from the preceding paragraphs that L generates a Markov semigroup □

In view of Theorem 3.3, we make the following

Definition 3.4. A semielliptic operator is an operator L ∈ D(A, ρ) satisfying

σ_{L}(ω^{∗}ω) ≤ 0, for every ω ∈ Ω^{1}.

The classification problem for Markov semigroups is now reduced to the problem of classifying semielliptic operators up to the natural notion of conjugacy: L ∈ D(A, ρ) and L′ ∈ D(A′, ρ′) are said to be conjugate if there is a ∗-isomorphism θ : A → A′ satisfying

ρ′ ∘ θ = ρ
(3.5)  L′ ∘ θ = θ ∘ L.

4. Momentum. Let (A, ρ) be a matrix algebra endowed with a faithful state ρ. In this section we show how semielliptic operators are constructed from more basic structures.

Let p be an element of A satisfying p^{∗} = −p. If we write D_{p}(a) = [p, a] = pa−ap, then D_{p} is a symmetric derivation of A; replacing p with p − ρ(p)1 if necessary, we can assume that p is normalized so that ρ(p) = 0. With this convention for normalization, the operator p is uniquely determined by the derivation D_{p}. Letting τ be the tracial state of A, we can define the density matrix h of ρ by ρ(a) = τ(ha), a ∈ A. h is a self adjoint matrix with strictly positive spectrum. Notice, finally, that ρ ∘ D_{p} = 0 iff p commutes with h; and in that case we have D_{p} ∈ D(A, ρ).

Definition 4.1. A momentum space is a pair (P, ⟨, ⟩) consisting of a real linear space P of skew-adjoint operators p ∈ A satisfying ρ(p) = 0 and ρ∘D_{p} = 0, together with a real inner product ⟨, ⟩ : P × P → ℝ.

We emphasize that the inner product on P can be specified arbitrarily, and in particular if A is realized concretely as B(H) for a finite dimensional Hilbert space H then there need be no relation between the inner products on P and H. Given a momentum space (P, ⟨, ⟩), we can construct an operator Δ ∈ D(A, ρ) as follows. Choose an orthonormal basis p_{1}, p_{2}, …, p_{r} for P and let D_{k}(a) = [p_{k}, a], 1 ≤ k ≤ r. Δ is defined by

(4.2)  Δ = ∑_{k=1}^{r} D_{k}^{2}.

Proposition 4.3. The operator Δ of (4.2) does not depend on the orthonormal basis chosen for P, and its symbol obeys

(4.3.1)  σ_{Δ}(dx dy) = −2ρ(Δ(x)y),

for all x, y ∈ A.

proof. Since D_{k} is a derivation in D(A, ρ) we have

ρ(D_{k}^{2}(x)y) = ρ(D_{k}(D_{k}(x)y)) − ρ(D_{k}(x)D_{k}(y)) = −ρ(D_{k}(x)D_{k}(y))

for each k, so that by (2.10)

ρ(Δ(x)y) = − ∑_{k−1}^{r} ρ(D_{k}(x)D_{k}(y)) = −\frac{1}{2}σ_{Δ}(dx dy).

This establishes (4.3.1).

Let p′_{1}, …, p′_{r} be another orthonormal basis for P and set Δ′ = ∑_{k} D_{k}^{′2}, where D′_{k} = [p′_{k}, ·]. Since ρ is a faithful state it suffices to show that ρ(Δ(x)y) = ρ(Δ′(x)y) for all x, y ∈ A; and by (4.3.1) this will follow from the assertion

∑_{k=1}^{r} ρ(D′_{k}(x)D′_{k}(y)) = ∑_{k=1}^{r} ρ(D_{k}(x)D_{k}(y)),

for all x, y ∈ A. To prove the latter we make the substitution p′_{k} = ∑_{i} ⟨p′_{k}, p_{i}⟩ p_{i} in the expression D′_{k}(x)D′_{k}(y) = [p′_{k}, x][p′_{k}, y] to obtain

D′_{k}(x)D′_{k}(y) = ∑_{i,j=1}^{r} ⟨p′_{k}, p_{i}⟩ ⟨p′_{k}, p_{j}⟩ D_{i}(x)D_{j}(y).

When the right side is summed on k, we may use the orthonormality of {p_{k}} and {p′_{k}} in the form ∑_{k} ⟨p′_{k}, p_{i}⟩ ⟨p′_{k}, p_{j}⟩ = δ_{ij} to obtain

∑_{k=1}^{r} D′_{k}(x)D′_{k}(y) = ∑_{k=1}^{r} D_{k}(x)D_{k}(y),

and the claim follows □

The operator Δ of equation (4.2) is called the Laplacian of the momentum space (P, ⟨, ⟩). Finally, let v be any skew-adjoint element of A for which D_{v}(a) = [v, a] obeys ρ ∘ D_{v} = 0, and set

(4.3)  L(x) = Δ(x) + [v, x],  x ∈ A.

Proposition 4.4. The operator L of (4.3) is a semielliptic operator in D(A, ρ).

proof. It is obvious that L satisfies the criteria (2.7) for membership in D(A, ρ).

Since D_{v} is a derivation we have σ_{L} = σ_{Δ}, and thus it suffices to show that Δ is semielliptic. For that, choose

ω = ∑_{k=1}^{n} a_{k} dx_{k}

in Ω^{1}. Then

ω^{∗}ω = ∑_{i,j=1}^{n}(dx_{i})^{∗} a_{i}^{∗}a_{j} dx_{j} = − ∑_{i,j=1}^{n} dx_{i}^{∗} a_{i}^{∗}a_{j} dx_{j}.

Thus by (2.11) we have

σ_{L}(ω^{∗}ω) = − ∑_{k=1}^{r} ∑_{i,j=1}^{n} ρ(D_{k}(x_{i}^{∗})a_{i}^{∗}a_{j}D_{k}(x_{j})) = − ∑_{k=1}^{r} ρ(z_{k}^{∗}z_{k}) ≤ 0,

where z_{k} = ∑_{j} a_{j}D_{k}(x_{j}), using the fact that D_{k}(x^{∗}) = D_{k}(x)^{∗} □

The following observation shows that an operator of the form (4.3) determines both of its summands uniquely. L^{2}(A, ρ) denotes the finite dimensional Hilbert space obtained by endowing A with the inner product ⟨a, b⟩_{ρ} = ρ(b^{∗}a).

Proposition 4.5. Let L = Δ + [v, ·] have the form (4.3), and consider L as an operator on the Hilbert space L^{2}(A, ρ). Then L + L^{∗} = 2Δ and L − L^{∗} = 2[v, ·].

proof. It suffices to show that ⟨Δ(a), b⟩_{ρ} = ⟨a, Δ(b)⟩_{ρ} and ⟨[v, a], b⟩_{ρ} = − ⟨a, [v, b]⟩_{ρ}. But if z is any element of A satisfying z^{∗} = −z and ρ([z, a]) = 0 for all a ∈ A, then the operator D(a) = [z, a] is a symmetric derivation which induces a skew-adjoint operator in B(L^{2}(A, ρ)). Indeed,

⟨D(a), b⟩_{ρ} = ρ(b^{∗}D(a)) = ρ(D(b^{∗}a)) − ρ(D(b^{∗})a) = 0 − ρ(D(b)^{∗}a) = − ⟨a, D(b)⟩_{ρ} .

Therefore D^{2} is a self-adjoint operator on L^{2}(A, ρ). It follows that Δ is a self-adjoint operator on L^{2}(A, ρ) and [v, ·] is skew-adjoint. (4.5) follows □

5. Modular Cohomology. The purpose of this section is to discuss certain cohomological issues so as to provide a context for the following section.

Definition 5.1. A modular algebra is a pair (A, ρ) consisting of a unital ∗-algebra A and a faithful state ρ on A with the following property: for every element a ∈ A there is an element δ(a) ∈ A such that

(5.1.1)  ρ(ab) = ρ(bδ(a)),  b ∈ A.

Remarks. By a faithful state on A we mean a linear functional ρ satisfying ρ(1) = 1 and ρ(a^{∗}a) > 0 for every a ≠ 0 ∈ A. Since ρ is faithful the bilinear form

a, b ↦ ρ(ab)

is nondegenerate, and hence the element δ(a) defined by (5.1.1) is unique.

Straightforward calculations show that δ is an automorphism of the algebra structure of A for which ρ ∘ δ = ρ, and from the property ρ(x^{∗}) = \overline{ρ(x)} one readily deduces

(5.2)  δ(a)^{∗} = δ^{−1}(a^{∗}),

δ^{−1} denoting the inverse automorphism. δ is called the modular automorphism of (A, ρ). We deviate from the traditional notation Δ for the modular automorphism associated with a faithful normal state of a von Neumann algebra in order to reserve Δ for the Laplacian of a momentum space.

Example 5.3. For our immediate purposes we have the case where A is the ∗- algebra M_{n}(ℂ) of n × n complex matrices and ρ is a faithful state. If τ is the normalized trace on M_{n}(ℂ) then we can define an operator h ∈ M_{n}(ℂ) by ρ(a) = τ(ha), a ∈ A. h is a self-adjoint matrix with strictly positive spectrum and δ is the inner automorphism δ(a) = hah^{−1}.

Example 5.4. Let M be an arbitrary von Neumann algebra and let ρ be a faithful normal state of M. Let σ^{ρ} be the modular automorphism group of ρ. We define A to be the set of all elements a ∈ M whose spectrum relative to this group is compact; equivalently, A consists of all elements a for which there is a function f = f_{a} ∈ L^{1}(ℝ) whose Fourier transform has compact support and for which

∫_{−∞}^{+∞} f(t) σ_{t}^{ρ}(a) dt = 0.

A is a unital ∗-algebra for which σ_{t}^{ρ}(A) = A for every t ∈ ℝ. For each a ∈ A, the function t ∈ ℝ ↦ σ_{t}^{ρ}(a) extends uniquely to an entire function

z ∈ ℂ ↦ σ_{z}^{ρ}(a) ∈ A.

{σ_{z}^{ρ} : z ∈ ℂ} defines a group of automorphisms of A parameterized by the additive group of complex numbers, satisfying σ_{z}^{ρ}(a)^{∗} = σρ ; \overline{z}(a^{∗}). The KMS condition as formulated in [23, 8.12.2] implies that the automorphism δ = σ_{i} satisfies

ρ(ab) = ρ(bδ(a)), a, b ∈ A.

Therefore (A, ρ) is a modular algebra.

Example 5.4 shows that the considerations of this section are appropriate in situations where there is no trace whatsoever, such as that in which one has an E_{0}-semigroup acting on a factor of type III. We intend to take up these more general issues in a subsequent paper.

We first sketch a generalization of Connes’ cyclic cohomology that is appropriate for modular algebras (A, ρ). By a modular cochain of dimension n ≥ 0 we mean a multilinear functional of n + 1 variables ϕ : A^{n+1} → ℂ satisfying

(5.5)  ϕ(a^{0}, a^{1}, …, a^{n}) = (−1)^{n}ϕ(δ^{−1}(a^{n}), a^{0}, …, a^{n−1}).

The formula (5.5) is ambiguous in the case n = 0, and in that case we intend that ϕ should satisfy ϕ(a) = ϕ(δ^{−1}(a)). C^{n}(A) denotes the vector space of n-dimensional cochains. For every ϕ ∈ C^{n}(A) we define a coboundary b_{δ}ϕ by

(5.6)  b_{δ}ϕ(a^{0}, …, a^{n+1}) = ∑_{k=0}^{n}(−1)^{k}ϕ(a^{0}, …, a^{k}a^{k+1}, …, a^{n+1})
+ (−1)^{n+1}ϕ(δ^{−1}(a^{n+1})a^{0}, a^{1}, …, a^{n}).

One can show that the operator b_{δ} maps C^{n}(A) into C^{n+1}(A) and satisfies b_{δ}^{2} = 0. The cohomology of the resulting complex is called modular cohomology. A multilinear form ϕ : A^{n+1} → ℂ satisfying (5.5) and b_{δ}ϕ = 0 is called a modular cocycle. ϕ is called exact if there is an n − 1 cochain ψ ∈ C^{n−1}(A) such that ϕ = b_{δ}ψ.

Notice that if ρ is a trace on A then δ is the identity automorphism and we have cyclic cohomology ([9, pp. 182–190]). In the case where ρ is a faithful state on a matrix algebra the resulting modular cohomology is trivial; nevertheless these considerations (especially for dimensions zero, one and two) will be relevant for our purposes below.

A zero cochain is a linear functional ϕ ∈ A′ satisfying ϕ ∘ δ = ϕ. Because ρ is faithful there is a unique operator p ∈ A for which ϕ(a) = ρ(ap), and the condition ϕ ∘ δ = ϕ is equivalent to δ(p) = p; i.e., p commutes with the “density matrix” of ρ.

A zero cocycle is simply a scalar multiple of ρ, ϕ(a) = λρ(a) for some λ ∈ ℂ. We will require the following simple characterization of exactness for one-cochains.

Proposition 5.7. Let ϕ ∈ C^{1}(A). Then ϕ is exact iff it has the form

(5.7.1)  ϕ(a, x) = ρ(a[v, x]),

where v is an element of A satisfying ρ([v, a]) = 0 for every a ∈ A.

proof. Supose first that p ∈ A satisfies ρ([p, a]) = 0 for all a ∈ A (equivalently, p commutes with the density matrix of ρ) and consider the form ϕ(a, x) = ρ(a[p, x]). Let ψ be the linear functional ψ(z) = −ρ(zp). Since δ(p) = p we have

ψ(δ(z)) = −ρ(δ(z)p) = −ρ ∘ δ(zp) = −ρ(zp) = ψ(z),

and hence ψ ∈ C^{0}(A). The coboundary of ψ is given by

b_{δ}ψ(a, x) = ψ(ax) − ψ(δ^{−1}(x)a) = −ρ(axp) + ρ(δ^{−1}(x)ap)
= ρ(apx) − ρ(axp) = ρ(a[p, x]),

and hence ϕ = b_{δ}ψ.

Conversely, suppose ϕ is exact and choose ψ ∈ C^{0}(A) such that ϕ(a, x) = ψ(ax)− ψ(δ^{−1}(x)a). Since ρ is faithful there is an element p ∈ A for which ψ(z) = −ρ(zp). One has δ(p) = p, and the preceding argument can be reversed to obtain formula (5.7.1) □

6. Exactness. A central issue in this work has been to appropriately characterize the semielliptic operators L ∈ D(A, ρ) that can be decomposed as in (4.3)

L = Δ + [v, ·]

as a first order perturbation of the Laplacian of some momentum space (P, ⟨·, ·⟩). In this section we introduce the notion of exact differential operator and give several characterizations of exactness (Theorem 6.9). In section 7 we show that the exact semielliptic operators are precisely those that can be so decomposed.

Throughout this section, (A, ρ) will denote a matrix algebra A endowed with a faithful state ρ. Let L ∈ D(A, ρ) be a differential operator. We define ω_{L} as the following bilinear form on A,

ω_{L}(x, y) = ρ(xL(y)) − ρ(L(x)y), x, y ∈ A.

ω_{L} will occupy a central role in the remainder of this paper.

Any operator L ∈ L(A) can be regarded as an operator on the Hilbert space L^{2}(A, ρ) and as such it has an adjoint L^{∗} ∈ B(L^{2}(A, ρ)). If L is a symmetric operator (i.e., L(x^{∗}) = L(x)^{∗} for every x ∈ A) then L^{∗} is uniquely defined by

(6.1)  ρ(L(x)y) = ρ(xL^{∗}(y)),  x, y ∈ A.

Notice that (6.1) is false if L is not symmetric, but it can be repaired in that case by replacing L(x) on the left side with L(x^{∗})^{∗}.

We are interested in operators that belong to D(A, ρ). For such an L we find that L^{∗}(1) = 0 by setting y = 1 in (6.1) and using ρ ∘ L = 0. Similarly, ρ ∘ L^{∗} = 0 follows from the condition L(1) = 0. However, the adjoint of an operator in D(A, ρ) need not symmetric. That information is contained in the form ω_{L} as follows.

Proposition 6.2. Let L ∈ D(A, ρ). Then L^{∗} ∈ D(A, ρ) if and only if ω_{L} is a modular one-cochain.

proof. The condition on ω_{L} means

(6.3)  ω_{L}(x, y) = −ω_{L}(δ^{−1}(y), x),  x, y ∈ A.

The left side of (6.3) is ρ(xL(y)) − ρ(L(x)y) while the right side is

−ρ(δ^{−1}(y)L(x)) + ρ(L(δ^{−1}(y))x) = −ρ(L(x)y) + ρ(δ^{−1}(y)L^{∗}(x))
= −ρ(L(x)y) + ρ(L^{∗}(x)y).

Setting these two expressions equal we find that (6.3) is equivalent to

(6.4)  ρ(xL(y)) = ρ(L^{∗}(x)y).

The left side of (6.4) can be rewritten as follows,

ρ(xL(y)) = \overline{ρ(L(y)^{∗}x^{∗})} = \overline{⟨x^{∗}, L(y)⟩}_{ρ} = \overline{⟨L^{∗}(x^{∗}), y⟩}_{ρ}
= \overline{ρ(y^{∗}L^{∗}(x^{∗}))} = ρ(L^{∗}(x^{∗})^{∗}y).

Thus (6.4) is equivalent to the formula L^{∗}(x^{∗})^{∗} = L^{∗}(x), i.e., that L^{∗} should be a symmetric operator □ Definition 6.5. An operator L ∈ D(A, ρ) is called exact if there is a modular zero-cochain ϕ ∈ C^{0}(A) such that ω_{L} = b_{δ}ϕ.

Remarks. Since the coboundary operator b_{δ} maps cochains to cochains, we see from Proposition 6.2 that for any exact operator L ∈ D(A, ρ) we have L^{∗} ∈ D(A, ρ). Exactness is considerably stronger, however, and the remainder of this section is devoted to giving several more concrete characterizations of exact differential operators.

We have already pointed out that the modular cohomology of a matrix algebra is trivial. Hence L is exact iff ω_{L} is a modular one-cocycle. Equivalently, an operator L ∈ D(A, ρ) is exact iff the following two conditions are satisfied

L^{∗} ∈ D(A, ρ) (6.6.1) b_{δ}ω_{L} = 0. (6.6.2)

Let δ be the modular automorphism of (A, ρ). δ induces a natural linear isomorphism δ̂: Ω^{1} → Ω^{1} defined by

(6.7)  δ̂(∑_{j} a_{j} dx_{j}) = ∑_{j} δ(a_{j}) dδ(x_{j}).

One has δ̂(aωb) = δ(a)δ̂(ω)δ(b) for a, b ∈ A, ω ∈ Ω^{1}. Notice too that the formula δ(a)^{∗} = δ^{−1}(a^{∗}) for a ∈ A implies that

(6.8)  δ̂(ω^{∗}) = δ̂^{−1}(ω)^{∗},  ω ∈ Ω^{1},

where δ̂^{−1} in (6.8) denotes the inverse of δ̂ : Ω^{1} → Ω^{1}, defined by δ̂^{−1}(a dx) = δ^{−1}(a) dδ^{−1}(x), a, x ∈ A.

Theorem 6.9. For any differential operator L in D(A, ρ), the following are equivalent.

(i) L is exact.

(ii) There is an element v ∈ A satisfying v^{∗} = −v, δ(v) = v, and

ω_{L}(x, y) = ρ(x[v, y]), x, y ∈ A.

(iii) L − L^{∗} is a derivation.

(iv) The symbol of L satisfies the KMS condition

σ_{L}(ω_{1}ω_{2}) = σ_{L}(ω_{2}δ̂(ω_{1})), ω_{1}, ω_{2} ∈ Ω^{1}.

For the proof, we require the following two results about symbols.

Lemma 6.10. Let L ∈ L(A) be an arbitrary operator. The symbol of L obeys

σ_{L}(aξ) = σ_{L}(ξ δ(a)), a ∈ A, ξ ∈ Ω^{2}.

proof. Using the homomorphism of modules θ_{L} ∈ hom(Ω^{2}, A) of Lemma 2.5 and the fact that σ_{L} = ρ ∘ θ_{L} we have

σ_{L}(aξ) = ρ(θ_{L}(aξ)) = ρ(aθ_{L}(ξ)) = ρ(θ_{L}(ξ)δ(a))
= ρ(θ_{L}(ξ δ(a)) = σ_{L}(ξ δ(a)),

as asserted □ Lemma 6.11. Let L be an operator in D(A, ρ) and let L^{∗} ∈ B(L^{2}(A, ρ)) be its adjoint. Then we have

σ_{L}(ω_{1}ω_{2}) = σ_{L^{∗}}(ω_{2}δ̂(ω_{1})),

for all ω_{1}, ω_{2} ∈ Ω^{1}.

proof. We claim first that for every a, x, y ∈ A we have

(6.12)  σ_{L}(dy a dx) = σ_{L^{∗}}(a dx dδ(y)).

Indeed, using Lemma 2.5 and the remarks following it, the left side of (6.12) can be written

σ_{L}(dy a dx) = ρ ∘ θ_{L}(dy a dx) = ρ(L(yax) − yL(ax) − L(ya)x + yL(a)x)
= −ρ(yL(ax)) − ρ(L(ya)x) + ρ(yL(a)x).

On the other hand,

σ_{L^{∗}}(a dx dδ(y)) = ρ(aL^{∗}(xδ(y)) − axL^{∗}(δ(y)) − aL^{∗}(x)δ(y) + axL^{∗}(1)δ(y)).

Using L^{∗}(1) = 0 and the formula ρ(uL^{∗}(v)) = ρ(L(u)v) the right side of the preceding formula becomes

ρ(L(a)xδ(y)) − ρ(L(ax)δ(y)) − ρ(yaL^{∗}(x)) = ρ(yL(a)x) − ρ(yL(ax)) − ρ(L(ya)x).

It follows that

σ_{L}(dy a dx) − σ_{L^{∗}}(a dx dδ(y)) = 0,

as asserted in (6.12).

We may conclude that for all ω_{2} ∈ Ω^{1} we have

σ_{L}(dy ω_{2}) = σ_{L^{∗}}(ω_{2}δ̂(dy)).

Now for any b, y ∈ A we have by Lemma 6.10

σ_{L}(b dy ω_{2}) = σ_{L}(dy ω_{2} δ(b)),

and by (6.12) the right side is σ_{L^{∗}}(ω_{2} δ(b) dδ(y)). Thus for any ω_{1} which is a finite sum of elements of the form b dy we have

σ_{L}(ω_{1}ω_{2}) = σ_{L^{∗}}(ω_{2} δ̂(ω_{1})),

and 6.11 follows □

Turning now to the proof of 6.9, we show (ii) ⟹ (i) ⟹ (iii) ⟹ (iv) ⟹ (ii). The first of these implications is immediate from Proposition 5.7.

(i) ⟹ (iii). Proposition 5.7 implies that ω_{L}(x, y) = ρ(xD(y)) where D is a derivation. On the other hand, by (6.11) we have

ω(x, y) = ρ(xL(y)) − ρ(L(x)y) = ρ(x(L(y) − L^{∗}(y)).

Hence L − L^{∗} = D is plainly a derivation.

(iii) ⟹ (iv). Since L − L^{∗} is a first order differential operator its symbol must vanish, and hence σ_{L} − σ_{L^{∗}} = σ_{L−L^{∗}} = 0. Thus by Lemma 6.11 we have

σ_{L}(ω_{1}ω_{2}) = σ_{L^{∗}}(ω_{2}δ̂(ω_{1})) = σ_{L}(ω_{2}δ̂(ω_{1})),

as asserted.

(iv) ⟹ (ii). Assuming the KMS condition 6.9 (iv), the preceding argument can clearly be reversed to show that σ_{L−L^{∗}} = 0, hence D = L − L^{∗} is a first order differential operator. Since L(1) = L^{∗}(1) = 0 we have D(1) = 0 and thus D is a derivation.

Choose an operator v ∈ A such that D(x) = [v, x], x ∈ A. We claim that v + v^{∗} is a scalar. To see this we use the fact that D = L − L^{∗} is a skew-adjoint operator on L^{2}(A, ρ) which satisfies ρ ∘ D = 0. Setting \overline{D}(x) = D(x^{∗})^{∗} we find that \overline{D} and D^{∗} are related by

ρ(\overline{D}(a)b) = ρ(aD^{∗}(b)).

Since D^{∗} = −D and since ρ ∘ D = 0, we have

ρ(aD^{∗}(b)) = −ρ(aD(b)) = −ρ(D(ab)) + ρ(D(a)b) = ρ(D(a)b).

Hence \overline{D} = D. Using the fact that D = [v, ·] and \overline{D} = [−v^{∗}, ·] we conclude that [v + v^{∗}, a] = 0 for every a ∈ A, hence v + v^{∗} is a scalar.

By replacing v with v − λ1 for a suitable real number λ, we can assume that v^{∗} = −v; and the condition ρ∘[v, ·] = 0 implies that δ(v) = v, completing the proof of Theorem 6.9 □

Corollary 6.13. Let (P, ⟨·, ·⟩) be a momentum space with Laplacian Δ and let L ∈ D(A, ρ) have the form

L = Δ + [v, ·]

where v^{∗} = −v, δ(v) = v. Then L is exact.

proof. By Proposition 4.5, L − L^{∗} = 2[v, ·] is a derivation and thus L satisfies condition (iii) of Theorem 6.9 □

Remarks. We emphasize that the symbol σ_{L} of an operator L ∈ D(A, ρ) does not normally give rise to a modular cocycle. More precisely, let ϕ : A^{3} → ℂ be the trilinear form

ϕ(a, x, y) = σ_{L}(a dx dy).

A straightforward computation shows that

b_{δ}ϕ = 0.

However, ϕ does not satisfy the functional equation for two-cochains

ϕ(a, x, y) = ϕ(δ^{−1}(y), a, x)

except in rather special circumstances. We omit further discussion of the latter since it does not come to bear on the sequel.

For an semielliptic operator L ∈ D(A, ρ), the form ω_{L}(x, y) = ρ(xL(y)) − ρ(L(x)y) plays a role closely analogous to the “driving force” of a classical mechanical system. Indeed, in what follows, the condition 6.9 (ii) for exactness,

ω_{L}(x, y) = −ρ(x[y, v])

will occupy a position parallel to the hypothesis of classical mechanics

F = −grad V

that the driving force should be conservative.

Example 6.14. We conclude this section by describing some elementary examples of semielliptic operators that are not exact. Let (A, τ) be the pair consisting of a full matrix algebra M_{n}(ℂ), n ≥ 3, with normalized trace τ. Let α be a ∗-automorphism of A and consider the operator

(6.15)  L(x) = α(x) − x,  x ∈ A.

One may verify directly that L is the generator of the semigroup

ϕ_{t} = e^{−t} exp(tα), t ≥ 0

and that (ϕ, τ) is a Markov semigroup. Thus, L is a semielliptic operator in D(A, τ). The following result implies that operators of the form (6.15) are typically not exact.

Proposition. Assume that there is an abelian ∗-subalgebra N ⊆ A such that α(N) = N and α^{2} ↾_{N} is not the identity map of N. Then L is not exact.

proof. Let ω_{L} : A × A → ℂ be the form

ω_{L}(x, y) = τ(xL(y)) − τ(L(x)y).

We will show that there is no derivation D of A for which ω_{L}(x, y) = τ(xD(y)), x, y ∈ A. This means that L fails to satisfy condition (ii) of Theorem 6.9, hence L is not exact.

Using the fact that every automorphism of A preserves the trace, we have

ω_{L}(x, y) = τ(x(α(y) − y) − (α(x) − x)y) = τ(xα(y) − α(x)y)
= τ(x(α(y) − α^{−1}(y)).

Thus we have to show that the operator α − α^{−1} is not a derivation.

But α − α^{−1} leaves the abelian ∗-subalgebra N invariant, so if it is a derivation it must be 0 on N. This implies that α(x) = α^{−1}(x) for every x ∈ N, contradicting the hypothesis on α □

7. Classification of semielliptic operators.

We now take up the classification of semielliptic operators in D(A, ρ). This is based on the classification of metrics. A will denote a full matrix algebra.

Definition 7.1. A metric on A is a linear functional g : Ω^{2} → ℂ satisfying

g(ω^{∗}ω) ≥ 0, ω ∈ Ω^{1}.

A metric gives rise to a positive semidefinite inner product (·, ·) : Ω^{1} × Ω^{1} → ℂ by way of

(ω_{1}, ω_{2}) = g(ω_{2}^{∗}ω_{1}), ω_{1}, ω_{2} ∈ Ω^{1}

and we have

(7.2)  (aω_{1}, ω_{2}) = (ω_{1}, a^{∗}ω_{2}).

Conversely, any (semidefinite) inner product (·, ·) satisfying (7.2) arises in this way from a unique metric g : Ω^{2} → ℂ. We have found it more convenient to work with metrics qua linear functionals rather than with metrics qua inner products. Notice, for example, that the symbol of any semielliptic operator L defines a metric g by way of g = −σ_{L}.

Metrics can be concretely presented as follows. Let z_{1}, z_{2}, …, z_{r} ∈ A and set

(7.3)  g(a dx dy) = ∑_{k=1}^{r} ρ(a[z_{k}^{∗}, x][z_{k}, y]).

It is easily seen that g is a metric. Notice that this metric satisfies

(7.4)  g(aξ) = g(ξ δ(a)),  a ∈ A, ξ ∈ Ω^{2},

δ denoting the modular automorphism of ρ. Conversely, it can be shown that any metric satisfying (7.4) can be expressed in the form (7.3) for some set of elements z_{1}, z_{2}, …, z_{r} in A. Note too that the z_{k} do not need to be self-adjoint or skewadjoint, and they need not satisfy δ(z_{k}) = z_{k}.

In this section we consider metrics g which satisfy the KMS condition of Theorem 6.9 (iv),

(7.5)  g(ω_{1}ω_{2}) = g(ω_{2} δ̂(ω_{1})),  ω_{1}, ω_{2} ∈ Ω^{1}.

We prove that a KMS metric can be decomposed into a sum of the form (7.3) in which the operators z_{1}, …, z_{r} satisfy the additional conditions z_{k}^{∗} = z_{k} and δ(z_{k}) = z_{k} for all k, and that moreover there is a unique momentum space associated with g.

Theorem 7.6. Let g be a nonzero metric which satisfies the KMS condition (7.5). There is a linearly independent set of self-adjoint operators x_{1}, …, x_{n} ∈ A such that δ(x_{k}) = x_{k} and ρ(x_{k}) = 0 for every k, and for which

g(a dx dy) = ∑_{k=1}^{n} ρ(a[x_{k}, x][x_{k}, y])

for every a, x, y ∈ A.

If x′_{1}, …, x′_{m} is another finite set satisfying all of these conditions then m = n and there is a real orthogonal n × n matrix (u_{ij}) such that

x′_{k} = ∑_{j=1}^{n} u_{kj}x_{j}, 1 ≤ k ≤ n.

Remarks. Before giving the proof of Theorem 7.6 we discuss some immediate consequences.

Notice that by replacing x_{k} with p_{k} = \sqrt{−1} x_{k} we obtain a linearly independent set of skew-adjoint elements {p_{1}, …, p_{n}} satisfying ρ([p_{k}, a]) = ρ(p_{k}) = 0 for every a ∈ A such that

g(a dx dy) = − ∑_{k=1}^{n} ρ(a[p_{k}, x][p_{k}, y]),

and for which a similar uniqueness holds. Thus the real vector space P spanned by {p_{1}, …, p_{n}} is independent of the particular choice of p_{1}, …, p_{n}, as is the inner product defined on P by

⟨p_{i}, p_{j}⟩ = δ_{ij}.

We conclude that every KMS metric g is associated with a unique momentum space (P, ⟨·, ·⟩).

This association g ⇝ (P, ⟨·, ·⟩) of a momentum space with a KMS metric is in fact bijective. Indeed, if we let Δ be the Laplacian of (P, ⟨·, ·⟩) then from (2.10) we find that 2g = −σ_{Δ} is determined by Δ, and therefore by (P, ⟨·, ·⟩). Thus Theorem 7.6 implies the following

Corollary 1. The above association defines a bijective correspondence between the set of KMS metrics and the set of momentum spaces in (A, ρ).

Indeed, Theorem 7.6 can be restated in the following equivalent (and somewhat more invariant) way.

Theorem 7.6A. For every KMS metric g : Ω^{2} → ℂ there is a unique momentum space (P, ⟨·, ·⟩) such that 2g = −σ_{Δ}, Δ denoting the Laplacian of (P, ⟨·, ·⟩).

Two metrics g_{1}, g_{2} are said to be equivalent if there is a ∗-automorphism α : A → A such that g_{2}(ξ) = g_{1}(α̂(ξ)), ξ ∈ Ω^{2}, α̂ denoting the induced mapping of Ω^{2}:

α̂(a dx dy) = α(a) dα(x) dα(y).

The above remarks lead immediately to the following classification of KMS metrics in terms of their momentum spaces.

Corollary 2. Let g_{1}, g_{2} be two KMS metrics with respective momentum spaces (P_{1}, ⟨·, ·⟩_{1}) and (P_{2}, ⟨·, ·⟩_{2}). g_{1} and g_{2} are equivalent iff there is a unitary operator u ∈ A such that

uP_{1}u^{∗} = P_{2},  and
⟨upu^{∗}, uqu^{∗}⟩_{2} = ⟨p, q⟩_{1} ,  p, q ∈ P_{1}.

proof. Assuming that g_{1} and g_{2} are equivalent, let α be a ∗-automorphism of A such that g_{2} = g_{1} ∘ α̂. We may find a unitary operator u ∈ A such that α(x) = uxu^{∗}. We can obviously define a momentum space (uP_{1}u^{∗}, ⟨·, ·⟩′) where ⟨upu^{∗}, uqu^{∗}⟩′ = ⟨p, q⟩_{1}, p, q ∈ P_{1}; and (uP_{1}u^{∗}, ⟨·, ·⟩′) gives rise to the metric g_{1} ∘ α̂ = g_{2}. Hence by the uniqueness assertion of Theorem 7.6 we must have (P_{2}, ⟨·, ·⟩) = (uP_{1}u^{∗}, ⟨·, ·⟩′).

The opposite implication is equally straightforward □

proof of Theorem 7.6. Since g(ω^{∗}ω) ≥ 0 for every ω ∈ Ω^{1}, a standard construction leads to a finite dimensional Hilbert space H_{g} and a complex linear map θ : Ω^{1} → H_{g} satisfying

(7.7.1)  θ_{g}(Ω^{1}) = H_{g},  and
(7.7.2)  g(ω_{2}^{∗}ω_{1}) = ⟨θ(ω_{1}), θ(ω_{2})⟩ ,  ω_{1}, ω_{2} ∈ Ω^{1}.

Let a be an element of A. The formula g(ω_{2}^{∗}aω_{1}) = g((a^{∗}ω_{2})^{∗}ω_{1}) implies that

⟨θ(aω_{1}), θ(ω_{2})⟩ = ⟨θ(ω_{1}), θ(a^{∗}ω_{2})⟩ ω_{i} ∈ Ω^{1}.

It follows that there is a unique ∗-representation π : A → B(H_{g}) defined by

(7.8)  π(a)θ(ω) = θ(aω),  ω ∈ Ω^{1}.

π(1) = 1 because 1 · ω = ω for every ω ∈ Ω^{1}.

Next, we show that there is a natural conjugation J of H_{g}, that is, an antilinear operator on H_{g} satisfying J^{2} = 1 and ⟨Jξ, Jη⟩ = ⟨η, ξ⟩, for all ξ, η ∈ H_{g}. For that we require some information about the behavior of g with respect to the (complex) modular automorphism group of the distinguished state on A, ρ(a) = trace(ha). For z ∈ ℂ there is an automorphism σ_{z} of the algebra structure of A defined by

σ_{z}(a) = h^{−iz}ah^{iz}.

We have σ_{z}σ_{w} = σ_{z+w} and σ_{z}(a^{∗}) = σz̄(a)^{∗}. For fixed a ∈ A, z ↦ σ_{z}(a) defines an entire function from ℂ to the Banach space A which is uniformly bounded on horizontal strips −M ≤ ℑ(z) ≤ M, 0 < M < ∞. For real z, σ_{z} is a ∗-automorphism of A.

Each automorphism σ_{z} determines a natural automorphism σ̂_{z} of the differential algebra Ω^{∗}, and in particular σ̂_{z} acts on Ω^{1} and Ω^{2} by

σ̂_{z}(a dx) = σ_{z}(a) dσ_{z}(x),
σ̂_{z}(a dx dy) = σ_{z}(a) dσ_{z}(x) dσ_{z}(y).

If z = t is real then σ̂_{t}(ω^{∗}) = σ̂_{t}(ω)^{∗} for all ω. The modular automorphism δ introduced in §5 is given by δ(a) = σ_{i}(a), a ∈ A.

We now define a second involution # of Ω^{1} using the natural square root of the modular automorphism

δ^{1/2}(a) = σ_{i/2}(a) = h^{1/2}ah^{−1/2}

as follows:

(7.9)  (a dx)^{#} = δ̂^{1/2}((a dx)^{∗}) = −dδ^{1/2}(x^{∗}) δ^{1/2}(a^{∗}).

Since δ^{1/2}(x^{∗}) = δ^{−1/2}(x)^{∗} we see that ω ↦ ω^{#} defines an involution in Ω^{1}.

Lemma 7.10. Every KMS metric g : Ω^{2} → ℂ satisfies

(i)  g(σ̂_{z}(ξ)) = g(ξ),  ξ ∈ Ω^{1}, z ∈ ℂ
(ii)  g(ω_{2}^{∗}ω_{1}) = g((ω_{1}^{#})^{∗}ω_{2}^{#})  ω_{1}, ω_{2} ∈ Ω^{1},

We have (aω)^{#} = ω^{#}δ^{1/2}(a^{∗}) and (ωb)^{#} = δ^{1/2}(b^{∗})ω^{#}, for every a, b ∈ A, ω ∈ Ω^{1}.

proof.

For (i), notice first that g ∘δ̂ = g. Indeed, for ω_{1}, ω_{2} ∈ Ω^{1} we can apply the KMS condition twice to obtain

g(δ̂(ω_{1}ω_{2})) = g(δ̂(ω_{1})δ̂(ω_{2})) = g(ω_{2}δ̂(ω_{1})) = g(ω_{1}ω_{2}),

and g ∘ δ̂ = g follows because Ω^{2} is spanned by products of the form ω_{1}ω_{2}, ω_{k} ∈ Ω^{1}.

Fix an element ξ ∈ Ω^{2} and consider the entire function

f(z) = g(σ̂_{z}(ξ)), z ∈ ℂ.

By the preceding remarks f is entire and bounded on the horizontal strip 0 ≤ ℑ(z) ≤ 1. We claim that f(z + i) = f(z) for every z ∈ ℂ. Indeed, since σ_{z+i} = σ_{i}σ_{z} = δσ_{z} and since g ∘ δ̂ = g we have

f(z + i) = g(δ̂(σ̂_{z}(ξ))) = g(σ̂_{z}(ξ)) = f(z).

Thus f is a bounded entire function which, by Liouville’s theorem, must be a constant.

For (ii) we write

g((ω_{1}^{#})^{∗}ω_{2}^{#}) = g((δ̂^{1/2}(ω_{1}^{∗}))^{∗}δ̂^{1/2}(ω_{2}^{∗})) = g(δ̂^{−1/2}(ω_{1})δ̂^{1/2}(ω_{2}^{∗})).

By what was just proved we can use g ∘ δ̂^{1/2} = g on the right hand term to obtain g(ω_{1}δ(ω_{2}^{∗})) = g(ω_{2}^{∗}ω_{1}),

and (ii) follows. The formula (aω)^{#} = ω^{#}δ^{1/2}(a^{∗}) follows directly from the definition of ^{#}. □

Lemma 7.10 (ii) implies that

⟨θ(ω_{1}), θ(ω_{2})⟩ = ⟨θ(ω_{2}^{#}), θ(ω_{1}^{#})⟩ ,

and hence we can define a unique antilinear isometry J of H_{g} by

(7.11)  Jθ(ω) = θ(ω^{#}),  ω ∈ Ω^{1}.

Since ω^{##} = ω it follows that J^{2} = 1, and hence J is a conjugation of H_{g}. We now define a unital ∗-antirepresentation π^{∘} : A → B(H_{g}) by

π^{∘}(a) = Jπ(a^{∗})J.

From the definitions of π and J we find that

(7.12)  π^{∘}(a)θ(ω) = θ(ω δ^{1/2}(a)),  a ∈ A, ω ∈ Ω^{1},

and hence π(a)π^{∘}(b) = π^{∘}(b)π(a) for every a, b ∈ A. Thus we have a ∗-representation of the C^{∗}-algebra A ⊗ A^{o} (A^{o} denoting the C^{∗}-algebra opposite to A) by way of

π ⊗ π^{∘} : a ⊗ b ∈ A ⊗ A^{o} ↦ π(a)π^{∘}(b).

Finally, the action of the modular group σ̂_{t} : Ω^{1} → Ω^{1} can be implemented by a one-parameter group of unitary operators U = {U_{t} : t ∈ ℝ}, which is defined on H_{g} as follows:

U_{t}θ(ω) = θ(σ̂_{t}(ω)), t ∈ ℝ, ω ∈ Ω^{1}.

Indeed, for ω_{1}, ω_{2} ∈ Ω^{1},

⟨θ(σ̂_{t}(ω_{1})), θ(σ̂_{t}(ω_{2})⟩ = g((σ̂_{t}(ω_{2}))^{∗}σ̂_{t}(ω_{1})) = g((σ̂_{t}(ω_{2}^{∗})σ̂_{t}(ω_{1}))
= g(σ̂_{t}(ω_{2}^{∗}ω_{1})) = g(ω_{2}^{∗}ω_{1}) = ⟨θ(ω_{1}), θ(ω_{2})⟩ ,

and thus each U_{t} is well-defined by the above formula. U obeys the group property and is strongly continuous.

Lemma 7.13. We may assume that H_{g} is the Hilbert space A^{n} consisting of all n-tuples z̄ = (z_{1}, …, z_{n}), z_{k} ∈ A, with inner product

⟨z̄, w̄⟩ = ∑_{k=1}^{n} ρ(w_{k}^{∗}z_{k}),

and that there is a set of n real numbers λ_{1}, …, λ_{n} ∈ ℝ such that the quadruple π, π^{∘}, U, J acts in the following way

(7.13.1)  π(a)z̄ = (az_{1}, …, az_{n})
(7.13.2)  π^{∘}(b)z̄ = (z_{1}δ^{1/2}(b), …, z_{n}δ^{1/2}(b)).
(7.13.3)  U_{t}z̄ = (e^{iλ_{1}t}σ_{t}(z_{1}), …, e^{iλ_{n}t}σ_{t}(z_{n})).
(7.13.4)  Jz̄ = (δ^{1/2}(z_{1}^{∗}), …, δ^{1/2}(z_{n}^{∗})),

Remark. We point out that in the course of the argument below, we will eventually prove that λ_{1} = ⋯ = λ_{n} = 0.

proof. Since A⊗A^{o} is another matrix algebra the commutant of the set of operators π(A) ∪ π^{∘}(A^{o}) is a type I_{n} factor for some n = 1, 2, …, and the representation π ⊗ π^{∘} is unitarily equivalent to a direct sum of n copies of any irreducible representation of A ⊗ A^{o}.

The standard representation λ : A → B(L^{2}(A, ρ)) is defined by

λ(a)z = az, z ∈ A

and there is a conjugation J_{0} of L^{2}(A, ρ) defined by

J_{0}(z) = δ^{1/2}(z^{∗}).

One has J_{0}λ(A)J_{0} = λ(A)′. The corresponding antirepresentation is defined by

λ^{∘}(b) = J_{0}λ(b^{∗})J_{0},

and one computes that

λ^{∘}(b)z = zδ^{1/2}(b).

Thus λ ⊗ λ^{∘} defines an irreducible representation of A ⊗ A^{o} on the Hilbert space

L^{2}(A, ρ). Hence π ⊗ π^{∘} is unitarily equivalent to a direct sum of n copies of λ ⊗ λ^{∘}.

After this change in coordinates, we may assume that π and π^{∘} act on A^{n} as specified in (7.13.1) and (7.13.2).

We show next that we can also achieve (7.13.3). Let U^{∘}_{t} be the unitary group which acts on A^{n} by way of

U^{∘}_{t}z̄ = (σ_{t}(z_{1}), …, σ_{t}(z_{n})), t ∈ ℝ, z_{i} ∈ A.

Since both U_{t} and U^{∘}_{t} induce the same action on π(A) and π^{∘}(A), namely

U_{t}π(a)U_{t}^{∗} = U^{∘}_{t}π(a)U^{∘} _{t}^{∗} = π(σ_{t}(a)),
U_{t}π^{∘}(a)U_{t}^{∗} = U^{∘}_{t}π^{∘}(a)U^{∘} _{t}^{∗} = π^{∘}(σ_{t}(a))

it follows that W_{t} = U_{t}U^{∘} _{t}^{∗} commutes with all operators in π(A) ∪ π^{∘}(A), and we

have U_{t} = W_{t}U^{∘}_{t}.

Notice that W = {W_{t} : t ∈ ℝ} is a one-parameter unitary group in the commutant of π(A) ∪ π^{∘}(A). Indeed, since σ_{t}(a) = h^{−it}ah^{it} is an inner automorphism of

A for each t it follows that U^{∘}_{t} belongs to the von Neumann algebra generated by

π(A) ∪ π^{∘}(A). Thus

W_{s+t}U^{∘}_{s+t} = U_{s+t} = U_{s}U_{t} = W_{s}U^{∘}_{s}W_{t}U^{∘}_{t} = W_{s}W_{t}U^{∘}_{s}U^{∘}_{t} = W_{s}W_{t}U^{∘}_{s+t},

so that W_{s}W_{t} = W_{s+t}. By the spectral theorem there are mutually orthogonal minimal projections E_{1}, …, E_{n} in (π(A) ∪ π^{∘}(A))′ and real numbers λ_{1}, …, λ_{n} such that

W_{t} = ∑_{k=1}^{n} e^{iλ_{k}t}E_{k}.

Thus there is a unitary operator V in the commutant of π(A) ∪ π^{∘}(A) which brings the unitary group W = {W_{t} : t ∈ ℝ} into diagonal form relative to the coordinates given in (7.13.3). Conjugation by this operator V does not change π or π^{∘}, and thus we have achieved (7.13.1), (7.13.2) and (7.13.3) simultaneously.

Let J_{n} be the natural conjugation of A^{n}, defined by

J_{n}(z̄) = (δ^{1/2}(z_{1}^{∗}), …, δ^{1/2}(z_{n}^{∗})),

and let J be the conjugation defined by (7.11). It remains to show that there is a unitary operator W on A^{n} such that

(7.14)  W ∈ (π(A) ∪ π^{∘}(A) ∪ {U_{t} : t ∈ ℝ})′,  and
(7.15)  J = W J_{n}W^{−1}.

Once we have such a W, we may use it to bring J into the form (7.13.4) without disturbing what has already been achived in (7.13.1), (7.13.2) and (7.13.3).

W is defined as follows. The standard conjugation J_{n} of A^{n} satisfies

J_{n}(π(a^{∗})z̄) = J_{n}(a^{∗}z_{1}, …, a^{∗}z_{n}) = (δ^{1/2}(z_{1}^{∗}a), …, δ^{1/2}(z_{n}^{∗}a))

= (δ^{1/2}(z_{1})δ^{1/2}(^{∗}a), …, δ^{1/2}(z_{n})δ^{1/2}(^{∗}a)) = π^{∘}(a)J_{n}(z̄),

and hence

(7.16)  J_{n}π(a^{∗})J_{n} = π^{∘}(a).

Since J_{n}^{2} = 1 we also have

(7.17)  J_{n}π^{∘}(a)J_{n} = π(a^{∗}).

Since J also satisfies (7.16) and (7.17) it follows that the unitary operator V = JJ_{n} must commute with both operator algebras π(A) and π^{∘}(A), and we have J = V J_{n}.

Notice that V also commutes with the unitary group {U_{t} : t ∈ ℝ}. Indeed, from the definitions of U_{t} and J we have

JU_{t}θ(ω) = Jθ(σ̂_{t}ω) = θ((σ̂_{t}ω)^{#}).

But for real t we have

(σ̂_{t}ω)^{#} = σ̂_{i/2}((σ̂_{t}ω)^{∗}) = σ̂_{i/2}σ̂_{t}(ω^{∗}) = σ̂_{t}(σ̂_{i/2}(ω^{∗})) = σ̂_{t}(ω^{#}).

Hence the right side of the previous equation is

θ(σ̂_{t}(ω^{#})) = U_{t}θ(ω^{#}) = U_{t}Jθ(ω).

We conclude that JU_{t} = U_{t}J. On the other hand, from the representation of U_{t} achieved in (7.13.3) we find that

J_{n}U_{t}z̄ = J_{n}(e^{iλ_{1}t}σ_{t}(z_{1}), …, e^{iλ_{n}t}σ_{t}(z_{n}))
= (σ_{i/2}(e^{−iλ_{1}t}σ_{t}(z_{1}^{∗}), …, σ_{i/2}(e^{−iλ_{n}t}σ_{t}(z_{n}^{∗})
= (σ_{t}((e^{iλ_{1}t}σ_{t}(z_{1}))^{#}), …, σ_{t}((e^{iλ_{n}t}σ_{t}(z_{n}))^{#})) = U_{t}J_{n}z̄.

Thus U_{t} commutes with both J and J_{n}, and therefore it must commute with V = JJ_{n}.

We now show that any “reasonable” square root W of V has the properties (7.14) and (7.15). Indeed, let

V = ∑_{k=1}^{r} λ_{k}E_{k}

be the spectral decomposition of V, where λ_{1}, …, λ_{r} are the distinct eigenvalues of V and E_{1}, …, E_{r} are its minimal spectral projections. Set

W = ∑_{k=1}^{r} λ_{k}^{1/2}E_{k},

where λ_{k}^{1/2} denotes either square root of λ_{k}. It is clear that W commutes with both π(A) and π^{∘}(A), and W commutes with each U_{t} because it belongs to the C^{∗}-algebra generated by the operator V ∈ {U_{t} : t ∈ ℝ}′. Since (UJ_{n})^{2} = J^{2} = 1 it follows that J_{n}UJ_{n} = U^{−1}. Now the operator mapping

α : T ↦ J_{n}T^{∗}J_{n}

is a ∗-antiautomorphism of the von Neumann algebra B(A^{n}) which carries the abelian C^{∗}-algebra C^{∗}(U) generated by U onto itself in such a way that α(U) = U. It follows that α(T) = T for all T ∈ C^{∗}(U). In particular α(W) = W, i.e., J_{n}W^{−1}J_{n} = W, and therefore U = W^{2} = W J_{n}W^{−1}J_{n}. Finally, we see that J = UJ_{n} = W J_{n}W^{−1}, as required □

We now describe θ in terms of the coordinates of Lemma 7.13.

Lemma 7.18. Considering A^{n} as a bimodule over A in the usual way,

az̄b = (az_{1}b, …, az_{n}b),  z̄ ∈ A^{n},

we have θ(aωb) = aθ(ω)b, a, b ∈ A, ω ∈ Ω^{1}.

proof. We have θ(aω) = π(a)θ(ω) = aθ(ω), by (7.13.1). For the right action we have

θ(ωb) = Jθ((ωb)^{#}) = Jθ(δ^{1/2}(b^{∗})δ̂^{1/2}(ω^{∗})) = Jπ(δ^{1/2}(b^{∗}))θ(ω^{#})

= Jπ(δ^{−1/2}(b)^{∗})Jθ(ω) = π^{∘}(δ^{−1/2}(b))θ(ω).

Notice that π^{∘}(δ^{−1/2}(b)) gives the (untwisted) right module action of b, since by (7.13.2)

π^{∘}(δ^{−1/2}(b))z̄ = (z_{1}b, …, z_{n}b),

z̄ ∈ A^{n}. Hence θ(ωb) = θ(ω)b □

It follows immediately from Lemma 7.18 that x ↦ θ(dx) is a derivation of A into the bimodule A^{n}, and hence there are derivations D_{1}, …, D_{n} of A such that

θ(a dx) = (aD_{1}(x), …, aD_{n}(x)), a, x ∈ A.

Lemma 7.19. The derivations D_{1}, …, D_{n} are linearly independent.

proof. Suppose that λ_{1}, …, λ_{n} ∈ ℂ are such that

∑_{k=1}^{n} λ_{k}D_{k}(x) = 0, x ∈ A.

Then for every a, x ∈ A we have

∑_{k=1}^{n} λ_{k}aD_{k}(x) = 0.

It follows that the linear operator L : A^{n} → A defined by L(z̄) = ∑_{k} λ_{k}z_{k} satisfies L(θ(a dx)) = 0 for all a, x ∈ A; hence L = 0 on the range of θ. Since θ(Ω^{1}) = H_{g} = A^{n}, L must be 0 □

The remainder of the proof is devoted to showing that D_{k}(a)^{∗} = −D_{k}(a^{∗}) and ρ ∘ D_{k} = 0, 1 ≤ k ≤ n. This is done in two steps, the first of which is the following.

Lemma 7.20. The derivations D_{1}, …, D_{n} have the form D_{k}(a) = [x_{k}h^{−1/2}, a], where x_{k} = x_{k}^{∗} and h is the density matrix of the state ρ.

proof. Applying (7.13.3) to the n-tuple z̄ = (D_{1}(x), …, D_{n}(x)) and using Jθ(dx) = θ((dx)^{#}) = −θ(dδ^{1/2}(x^{∗})) we obtain

−D_{k}(δ^{1/2}(x^{∗})) = δ^{1/2}(D_{k}(x)^{∗}), 1 ≤ k ≤ n.

After choosing elements z_{k} ∈ A so that D_{k}(x) = [z_{k}, x] and unravelling the preceding formula, one finds that for all x ∈ A one has

−[z_{k}, δ^{1/2}(x^{∗})] = −[δ^{1/2}(z_{k}^{∗}), δ^{1/2}(x^{∗})].

Thus z_{k} − δ^{1/2}(z_{k}^{∗}) commutes with everything in A, and must be a scalar multiple of 1 for every 1 ≤ k ≤ n.

Now if z ∈ A and λ ∈ ℂ are such that z − δ^{1/2}(z^{∗}) = λ1, then by applying the normalized trace τ on A we find that λ = τ(z) − \overline{τ(z)} is imaginary. Writing λ = iλ_{0} where λ_{0} ∈ ℝ we can replace z with z_{0} = z − \frac{iλ_{0}}{2} 1 and we find that z_{0} − δ^{1/2}(z_{0}^{∗}) = 0. Noting that δ^{1/2}(x) = h^{1/2}xh^{−1/2}, the latter implies that x = z_{0}h^{−1/2} is self-adjoint, and hence

z = xh^{−1/2} + \frac{iλ_{0}}{2} 1.

The required representation of z_{1}, …, z_{n} follows □

Now by definition of U_{t} we have

U_{t}(D_{1}(x), …, D_{n}(x)) = U_{t}(θ(dx)) = θ(dσ_{t}(x)) = (D_{1}(σ_{t}(x)), …, D_{n}(σ_{t}(x))),

while from (7.13.3) we have

U_{t}(D_{1}(x), …, D_{n}(x)) = (e^{iλ_{1}t}σ_{t}(D_{1}(x)), …, e^{iλ_{n}t}σ_{t}(D_{n}(x)).

It follows that

σ_{t}(D_{k}(x)) = e^{−iλ_{k}t}D_{k}(σ_{t}(x)), 1 ≤ k ≤ n, t ∈ ℝ.

By (7.20), D_{k} has the form D_{k}(x) = [x_{k}h^{−1/2}, x], and thus

σ_{t}(x_{k}h^{−1/2}) − e^{−iλ_{k}t}x_{k}h^{−1/2}

must be a scalar. The following lemma shows that this implies that x_{k} must commute with h.

Lemma 7.21. Let λ ∈ ℝ and let x be a self adjoint element of A such that

σ_{t}(xh^{−1/2}) − e^{iλt}xh^{−1/2}

is a scalar for every t ∈ ℝ. Then λ = 0 and x commutes with h.

proof. For every t ∈ ℝ let μ(t) be the complex number defined by

σ_{t}(xh^{−1/2}) − e^{iλt}x_{k}h^{−1/2} = μ(t)1.

Applying the normalized trace τ of A to both sides of this equation we find that for c = τ(xh^{−1/2}) we have μ(t) = c(1 − e^{iλt}). Thus

σ_{t}(xh^{−1/2} − c1) = e^{iλt}(xh^{−1/2} − c1).

After multiplying through on the right by h^{1/2} we find that

(7.22)  σ_{t}(x − ch^{1/2}) = e^{iλt}(x − ch^{1/2}),  t ∈ ℝ.

Now c = τ(xh^{−1/2}) = τ(h^{−1/4}xh^{−1/4}) is a real number because x is a self-adjoint operator, and thus the operator x − ch^{1/2} appearing in (7.22) is self-adjoint. The left side of (7.22) is therefore a self-adjoint operator for every t ∈ ℝ, and hence (7.22) implies that λ = 0 except in the trivial case where x = ch^{1/2}.

In either case it follows that x − ch^{1/2} is fixed under the action of σ_{t}, t ∈ ℝ, and hence x must commute with h □

Applying Lemma 7.21 to the derivations D_{k} we find that D_{k}(a) = [x_{k}h^{−1/2}, a] = [h^{−1/4}x_{k}h^{−1/4}, a] is implemented by the self-adjoint operator y_{k} = h^{−1/4}x_{k}h^{−1/4}, where y_{k} commutes with h. By replacing y_{k} with y_{k} − ρ(y_{k})1 we can also assume that ρ(y_{k}) = 0. That completes the proof of all but the uniqueness assertion of Theorem 7.6.

For uniqueness, let x′_{1}, …, x′_{n} be a linearly independent set of self-adjoint operators satisfying δ(x′_{k}) = x′_{k}, ρ(x′_{k}) = 0, and

(7.23)  g(a dx dy) = ∑_{k=1}^{m} ρ(a[x′_{k}, x][x′_{k}, y])

for all a, x, y ∈ A. Define a linear map θ′ : Ω^{1} → A^{m} by

(7.24)  θ′(a dx) = (a[x′_{1}, x], …, a[x′_{m}, x]).

If we make A^{m} into a Hilbert space m · L^{2}(A, ρ) as in Lemma 7.13, and into a bimodule as in Lemma 7.18, then we find that

(7.25)  θ′(aωb) = aθ′(ω)b,  a, b ∈ A, ω ∈ Ω^{1}.

We claim that θ′(Ω^{1}) = A^{m}. For that, note that (7.25) implies that the subspace θ′(Ω^{1}) is invariant under left and right multiplication by elements of A, and hence the projection P onto the range of θ′ must have the form P = (p_{ij}1) where (p_{ij}) is an m × m matrix of complex scalars. If P ≠ 1 then 1 − P must have a nonzero row with entries μ_{1}, …, μ_{m}. Since (1 − P)θ′(dx) = 0 for every x ∈ A it follows that

μ_{1}[x′_{k}, x] + ⋯ + μ_{m}[x′_{m}, x] = 0

for every x ∈ A and hence ∑ μ_{k}x′_{k} must be a scalar multiple ν1 of 1. However, since ρ(x′_{k}) = 0 and ρ is a state, we must have ν = 0. Hence ∑ μ_{k}x′_{k} = 0, contradicting linear independence.

Now the formulas (7.23) and (7.24) imply that for all ω_{1}, ω_{2} ∈ Ω^{1},

⟨θ′(ω_{1}), θ′(ω_{2})⟩ = g(ω_{2}^{#}ω_{1}) = ⟨θ(ω_{1}), θ(ω_{2}))⟩ .

Thus we can define a unique unitary operator W : A^{n} → A^{m} by

W θ(ω) = θ′(ω), ω ∈ Ω^{1}.

Since W is unitary and both θ and θ′ are bimodule homomorphisms it follows that W must implements a ∗ isomorphism of the commutant of π(A)∪π^{∘}(A) onto that of π′(A) ∪ π^{∘}′(A). Since these commutants are factors of type I_{n} and I_{m} respectively, we conclude that m = n, and that W commutes with both the left and right module actions of A on A^{n}. It follows that there is an n × n scalar unitary matrix (w_{ij}) such that W = (w_{ij}1).

We claim that the scalars w_{ij} are real numbers, so that (w_{ij}) is a real orthogonal n×n matrix. Indeed, if J_{n} is the standard involution of A^{n} defined in (7.13.4) then we have

J_{n}θ(ω) = θ(ω^{#}),  and
J_{n}θ′(ω) = θ′(ω^{#}).

It follows that

W J_{n}θ(ω) = W θ(ω^{#}) = θ′(ω^{#}) = J_{n}θ′(ω) = J_{n}W θ(ω), ω ∈ Ω^{1},

and hence J_{n}W = W J_{n}. Noting that the action of J_{n} on scalar operator matrices obeys

J_{n}(λ_{ij}1)J_{n}^{−1} = (\overline{λ}_{ij}1),

we conclude that \overline{w}_{ij} = w_{ij} for all i, j, as asserted.

Finally, since θ′(dx) = W θ(dx), we have

[x′_{k}, x] = ∑_{j=1}^{n} w_{kj}[x_{j}, x] = [∑_{j} w_{kj}x_{j}, x]

for all x ∈ A it follows that each operator

x′_{k} − ∑_{k=1}^{n} w_{kj}x_{j}

must be a scalar multiple of 1. Since ρ(x_{k}) = ρ(x′_{k}) = 0 and ρ is a state, the scalars must be 0 and we have the asserted relation

x′_{k} = ∑_{k=1}^{b} w_{kj}x_{j}, 1 ≤ k ≤ n.

That completes the proof of Theorem 7.6 □

Theorem 7.6 is applied to the classification of semielliptic operators as follows.

Theorem 7.26. Let L ∈ D(A, ρ) be an exact semielliptic operator. Then there is a unique momentum space (P, ⟨·, ·⟩) and a unique skew-adjoint operator v such that δ(v) = v, ρ(v) = 0, and

(7.26.1)  L = Δ + [v, ·],

Δ denoting the Laplacian of (P, ⟨·, ·⟩).

Conversely, any operator L of the form (7.26.1) is an exact semielliptic operator.

proof. Since L is an exact semielliptic operator its symbol σ_{L} satisfies condition (iii) of Theorem 6.9, hence

2g = −σ_{L}

defines a KMS metric on Ω^{2}. By Theorem 7.6A there is a unique momentum space (P, ⟨·, ·⟩) whose Laplacian Δ satisfies

2g = −σ_{Δ}.

The symbol of L − Δ therefore vanishes, so that D = L − Δ is a symmetric derivation for which ρ ∘ D = 0. We can find a skew-adjoint operator v ∈ A for which D(x) = [v, x], and the condition ρ ∘ D = 0 implies that v commutes with the density matrix of ρ. Replacing v with v − ρ(v)1, we obtain (7.26.1).

The converse was established in Corollary 6.13.

Exact semielliptic operators are classified in terms of their momenta and potentials as follows.

Theorem 7.27. Let L_{1}, L_{2} be two exact semielliptic operators in D(A, ρ) with momentum spaces (P_{k}, ⟨·, ·⟩_{k}) and natural decompositions L_{k} = Δ_{k} + [v_{k}, ·] , k = 1, 2. Then L_{1} and L_{2} are conjugate iff there is a ∗-automorphism α of A such that ρ ∘ α = ρ, and

(7.27.1)  α(P_{1}) = P_{2},
(7.27.2) ⟨α(p), α(q)⟩_{2} = ⟨p, q⟩_{1} , p, q ∈ P_{1}
(7.27.3)  α(v_{1}) = v_{2}.

proof. Assume that L_{1} and L_{2} are conjugate. Thus there is a ∗-automorphism α of A such that ρ ∘ α = ρ and α ∘ L_{1} = L_{2} ∘ α. Since α preserves ρ, it induces a unitary operator on L^{2}(A, ρ) in the natural way:

U_{α} : x ↦ α(x), x ∈ A,

and we have U_{α}L_{1}U_{α}^{−1} = L_{2}. Hence U_{α}(L_{1}+L_{1}^{∗})U_{α}^{−1} = L_{2}+L_{2}^{∗}. Since Δ_{k} = L_{k}+L_{k}^{∗} it follows that αΔ_{1}α^{−1} = Δ_{2}. Hence α must induce an isomorphism of momentum spaces in the sense specified in (7.27.1) and (7.27.2).

Setting D_{k} = [v_{k}, ·], we have

α ∘ D_{1} ∘ α^{−1} = α ∘ (L_{1} − Δ_{1}) ∘ α^{−1} = L_{2} − Δ_{2} = D_{2},

so that for all x in A we have α([v_{i}, α^{−1}(x)] = [v_{2}, x]. The latter implies that α(v_{1}) − v_{2} is a scalar; and since ρ(v_{2}) = ρ(α(v_{1})) = ρ(v_{1}) = 0, we deduce (7.27.3).

The proof of the converse is similar □

Remarks. Of course, the ∗ automorphism α can be implemented by a unitary operator.

Notice too that the three invariants in (7.26.1)–(7.26.3) that classify exact semielliptic operators can be specified independently of one another. For example, starting with any such operator L one may scale the metric ⟨·, ·⟩ on P to obtain a family of mutually non-conjugate operators, or one can scale the potential v to obtain a second family of mutually non-conjugate operators.

It also makes sense to speak of semielliptic operators that are “free” in the sense that their potential operator v is zero.

8. Applications. The preceding results lead directly to a new classification of E_{0}-semigroups. For that one considers pairs (α, ω) consisting of an E_{0}-semigroup α = {α_{t} : t ≥ 0} acting on B(H_{α}) and a normal state ω of B(H_{α}) satisfying

ω ∘ α_{t} = ω, t ≥ 0.

Two such pairs (α, ω) and (α′, ω′) are said to be conjugate if there is a ∗-isomorphism θ : B(H_{α}) → B(H_{α′}) such that

(8.1.1)  θ ∘ α_{t} = α′ ∘ θ,  and
(8.1.2)  ω′ ∘ θ = ω.

Proposition 8.2. Assume that θ is a ∗-isomorphism satisfying (8.1.1) and (8.1.2) and let p_{0} (resp. p′_{0}) be the support projection of ω (resp. ω′). Let H_{0} = p_{0}H_{α}, H′_{0} = p′_{0}H_{α′} and let (ϕ, ρ), (ϕ′, ρ′) be the Markov semigroups obtained by compressing α, α′ to B(H_{0}), B(H′_{0}) as in §1:

ϕ_{t}(a) = p_{0}α_{t}(ap_{0})p_{0},
ϕ′_{t}(b) = p′_{o}α′_{t}(bp_{0})p′_{0},  t ≥ 0,

where ρ, ρ′ are obtained from ω, ω′ by restriction. Then the restriction θ_{0} of θ to B(H_{0}) implements a conjugacy of (ϕ, ρ) and (ϕ′, ρ′).

proof. The argument is straightforward once one observes that the condition ω′∘θ = ω implies that θ(p_{0}) = p′_{0} and ρ′ ∘ θ_{0} = ρ □

Definition 8.3. An E_{0}-semigroup pair (α, ω) is said to be of finite type if the support projection of ω is finite dimensional.

If (α, ω) is of finite type and (ϕ, ρ) is its natural Markov semigroup obtained by compression as in §1 then we may consider the generator L of ϕ,

(8.4)  ϕ_{t} = exp(tL),  t ≥ 0.

L is an semielliptic operator in D(B(H_{0}), ρ) and it is sensible to ask if L is exact.

Definition 8.5. (α, ω) is called exact if it is of finite type and the generator L of its associated Markov semigroup is an exact semielliptic operator.

Notice that if (α, ω) is exact then the generator L of (8.4) decomposes uniquely as in Theorem 7.6,

L = Δ + [v, ·]

where Δ is the Laplacian of a unique momentum space (P, ⟨·, ·⟩) and v is a skewadjoint operator satisfying ρ(v) = 0 and ρ∘[v, ·] = 0. The triple (P, ⟨·, ·⟩ , v) is called the dynamical invariant of (α, ω). It is truly an invariant because of the following result.

Theorem 8.6. If two exact E_{0}-semigroup pairs (α, ω) and (α′, ω′) are conjugate then their dynamical invariants (P, ⟨·, ·⟩ , v) and (P′, ⟨·, ·⟩′ , v′) are isomorphic in the sense that there is a ∗-isomorphism α : B(H_{0}) → B(H′_{0}) which satisfies the conditions of Theorem 7.27:

ω′ ∘ α = ω
α(P) = P′
⟨α(p), α(q)⟩′ = ⟨p, q⟩ ,  p, q ∈ P
α(v) = v′.

proof. Given the preceding remarks, the argument is a simple variation of what was done in the proof of Theorem 7.27 □

Remarks. We have already pointed out in §1 that the converse of Theorem 8.6 is false in general, but that it is true if both α and α′ are minimal. More generally, we can paraphrase a recent dilation theorem of B. V. R. Bhat [6] in our context as follows.

Theorem 8.7. Let (ϕ, ρ) be a Markov semigroup acting on a separable Hilbert space H_{0}. Then there is a minimal E_{0}-semigroup α acting on a Hilbert space H ⊇ H_{0} and a normal state ω on B(H) such that ω ∘ α_{t} = ω for every t ≥ 0 and such that (ϕ, ρ) is obtained from (α, ω) by compression as described above.

If (ϕ′, ρ′) is another Markov semigroup which is conjugate to (ϕ, ρ) and which gives rise to a minimal E_{0}-semigroup pair (α′, ω′) then (α, ω) and (α′, ω′) are conjugate.

Remark 8.8. Taken together, Theorems 8.6 and 8.7 imply that exact minimal E_{0}- semigroup pairs (α, ω) are completely classified up to conjugacy by their dynamical invariants (P, ⟨·, ·⟩ , v). More explicitly, suppose we start with a pair (A, ρ) consisting of a faithful state ρ on A = B(H_{0}), where H_{0} is a finite dimensional Hilbert space. Choose an arbitrary dynamical invariant (P, ⟨·, ·⟩ , v) in (A, ρ) and form its corresponding semielliptic operator L ∈ D(A, ρ):

L(x) = Δ(x) + [v, x], x ∈ A,

Δ being the Laplacian of (P, ⟨·, ·⟩). One exponentiates L to obtain ϕ_{t} = exp(tL) and a Markov semigroup (ϕ, ρ). Finally, one obtains a minimal E_{0}-semigroup pair (α, ω) by the dilation procedure of Theorem 8.7. Such an (α, ω) is an exact pair whose dynamical invariant is isomorphic to (P, ⟨·, ·⟩ , v). Every exact pair arises in this way, and two exact pairs are conjugate if and only if their corresponding dynamical invariants are isomorphic.

We infer from these remarks that the dynamical invariant (P, ⟨·, ·⟩ , v) consists of infinitesimal structures that completely determine the associated flow (α, ω). Thus dynamical invariants are analogous to the differential equations that govern the behavior of classical mechanical systems. There are obvious implications for the classification of type I primary histories.

Finally, it is possible to determine which triples (P, ⟨·, ·⟩ , v) lead to “mixing” properties for their associated E_{0}-semigroups. By that we mean that all limits of the form

lim_{t→∞} ϕ(α_{t}(a)), a ∈ B(H_{α}), ϕ ∈ B(H_{α})_{∗}

should exist. These issues will be taken up elsewhere.

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