INFINITE TENSOR PRODUCTS OF

COMPLETELY POSITIVE SEMIGROUPS

William Arveson and Geoffrey Price

Department of Mathematics University of California Berkeley CA 94720, USA

Department of Mathematics U. S. Naval Academy Annapolis, MD 21402, USA

Abstract. We construct a new class of semigroups of completely positive maps on B(H) which can be decomposed into an infinite tensor product of such semigroups. Under suitable hypotheses, the minimal dilations of these semigroups to E_{0}-semigroups are pure, and have no normal invariant states. Concrete examples are discussed in some detail.

Dedicated to Robert T. Powers on the occasion of his sixtieth birthday

1. Introduction.

An E_{0}-semigroup is a semigroup of normal unital ∗-endomorphisms α = {α_{t} : t ≥ 0} acting on the algebra B(H) of all bounded operators on a separable Hilbert space H, which satisfies the natural continuity condition

lim_{t→0^{+}}⟨α_{t}(x)ξ, η⟩ = ⟨xξ, η⟩, x ∈ B(H), ξ, η ∈ H.

E_{0}-semigroups are classified roughly into types I, II, III, depending on the existence of “intertwining semigroups”. In general, the von Neumann algebras N_{t} = α_{t}(B(H)) are type I subfactors of B(H) which decrease with increasing t. An E_{0}-semigroup α is called pure if its “tail” algebra is trivial in the sense that

(1.1)  ∩_{t≥0}α_{t}(B(H)) = ℂ1.

Pure E_{0}-semigroups emerge naturally in noncommutative dynamics. For example, Powers [P2] has developed a “standard form” for most E_{0}-semigroups (more precisely, those which are not of type III), which is analogous to the standard form for von Neumann algebras. A key element is the result that any E_{0}-semigroup not of type III can be perturbed by a cocycle so as to become a pure E_{0}-semigroup which leaves invariant a (necessarily unique) vector state. The perturbed semigroup is said to be in standard form. This standard form is unique in the sense that if two E_{0}-semigroups in standard form are cocycle-conjugate then they must actually be conjugate.

Key words and phrases. completely positive semigroup, minimal dilation, pure E_{0}-semigroup, absorbing state, invariant normal state.

The authors acknowledge support from NSF grants DMS-9802474 and DMS-9706441.

On the other hand, in the interaction theory developed by one of us [A7], one is presented with a pair of pure E_{0}-semigroups, each of which has an invariant normal state. Significantly, in the context of interactions the invariant states are typically not vector states. Given any pure E_{0}-semigroup α with an invariant normal state ω, an arbitrary normal state ρ of B(H) can be restricted to each of the type I subfactors α_{t}(B(H)), t ≥ 0, and each of these restrictions has an eigenvalue list. The basic interaction inequality of [A2] depends upon an asymptotic formula which relates the limiting behavior of these eigenvalue lists to those of ρ and ω as t → ∞.

Consideration of such issues has led us to single out pure E_{0}-semigroups for attention, and many significant problems remain open. For example, Powers’ result above implies that every E_{0}-semigroup that is not of type III is a cocycle perturbation of a pure E_{0}-semigroup, but it is not known if every E_{0}-semigroup can be perturbed by a cocycle into a pure one. In this paper we are concerned with the construction of pure E_{0}-semigroups which do not have normal invariant states. The first examples of this phenomenon were given in [A3], and were based on the canonical commutation relations. In this paper we give a new and more flexible construction of such examples. As in [A3], we actually construct examples of semigroups of completely positive linear maps of B(H) having appropriate properties, and then obtain the examples of E_{0}-semigroups by a dilation procedure. However, the examples constructed here differ from from those of [A3] in that they decompose into infinite tensor products of simpler flows acting on matrix algebras.

We now describe the setting of this paper in more detail. By a CP semigroup we mean a one-parameter semigroup ϕ = {ϕ_{t} : t ≥ 0} of normal completely positive linear maps of B(H) which is unital in the sense that ϕ_{t}(1) = 1 for every t, and is continuous in t as described above for E_{0}-semigroups. The definition of pure E_{0}-semigroup (1.1) must be modified appropriately for the broader category of CP semigroups. A CP semigroup ϕ is said to be pure if for every pair of normal states ρ, σ of B(H) we have

(1.2)  lim_{t→∞} ∥ρ ∘ ϕ_{t} − σ ∘ ϕ_{t}∥ = 0.

It is known that (1.1) and (1.2) are equivalent in case ϕ = α is an E_{0}-semigroup [A2]. Notice that if ϕ is a pure CP semigroup and if there is a normal state ω of B(H) that is ϕ-invariant in the sense that ω ∘ ϕ_{t} = ω for every t ≥ 0, then because of (1.2) ω must be an absorbing state in the sense that for every normal state ρ we have

(1.3)  lim_{t→∞} ∥ρ ∘ ϕ_{t} − ω∥ = 0.

When an absorbing state exists, it is obviously the unique normal ϕ-invariant state. Conversely, if for an arbitrary CP semigroup ϕ there is a state ω of B(H) satisfying (1.3) for every normal state ρ, then ω is a (normal) absorbing state and ϕ must be a pure CP semigroup.

In the physics literature the asymptotic behavior (1.3) is called return to equilibrium, whereas in ergodic theory it is called mixing. Indeed, (1.3) implies that for every pair of operators A, B ∈ B(H) and every normal state σ on B(H) one has

lim_{t→∞} σ(ϕ_{t}(A)B) = ω(A)σ(B),

and in particular ω is a mixing state: ω(ϕ_{t}(A)B) → ω(A)ω(B) as t → ∞.

In general, a pure CP semigroup need not have an absorbing state (equivalently, there may not exist a normal state that is ϕ-invariant). But in the context of CP semigroups acting on matrix algebras, (1.2) and (1.3) are equivalent. Indeed, if H is finite dimensional and ϕ is a pure CP semigroup acting on B(H), then a simple application of the Markov-Kakutani fixed point theorem shows that there must be a state ω of B(H) satisfying ω ∘ ϕ_{t} = ω for every t ≥ 0. Since in this case ω must be normal, the preceding observations imply that (1.3) is satisfied.

In section 2, we show how to construct examples of pure CP semigroups which decompose into an infinite tensor product of CP semigroups acting on matrix algebras (Theorem A). Under appropriate hypotheses, these CP semigroups do not have normal invariant states. In section 3 we discuss examples and show how to arrange the general hypotheses of Theorem A. These results are applied to the theory of E_{0}-semigroups in section 4.

2. Infinite tensor products of CP Semigroups.

In this section we discuss general problems associated with the construction of infinite tensor products of CP semigroups. We give an effective criterion for their existence, and we determine when such infinite tensor products are pure. Applications are taken up in the following section.

Starting with a sequence of CP semigroups ϕ^{k} acting on B(H_{k}), k = 1, 2, …, one can choose an arbitrary sequence of normal pure states ω_{k} of B(H_{k}), k = 1, 2, …, and with that data attempt to construct an infinite spatial tensor product ⊗_{k}ϕ^{k} of CP semigroups. One finds, however, that the existence of such a product semigroup depends strongly on the choice of the sequence ω_{1}, ω_{2}, . . .. In order to discuss this, we begin by introducing an intermediate C^{∗}-algebra A which is in some sense an infinite tensor product of type I factors, and which carries “locally” the structure of a von Neumann algebra. On A there is a natural way of defining the infinite tensor product of CP semigroups, but there is no Hilbert space. We show that for appropriate sequences (ω_{k}), the GNS construction applied to ⊗_{k}ω_{k} gives rise to a representation of A on a Hilbert space H so that the tensor product of semigroups can be extended uniquely to a CP semigroup on B(H).

Given a pair of normal completely positive linear maps ϕ, ψ on B(H), B(K) respectively, there is a unique normal completely positive linear map ϕ ⊗ ψ on B(H ⊗ K), defined uniquely by its action on operators A ⊗ B,

ϕ ⊗ ψ(A ⊗ B) = ϕ(A) ⊗ ψ(B), A ∈ B(H), B ∈ B(K).

Similarly, there is a natural notion of finite tensor product of normal completely positive linear maps; when ϕ^{k} acts on B(H_{k}), k = 1, …, n, ϕ^{1} ⊗⋯⊗ϕ^{n} is a normal completely positive linear map on B(H_{1} ⊗ ⋯ ⊗ H_{n}).

Suppose now that we have a sequence of CP semigroups ϕ^{k} = {ϕ_{t}^{k} : t ≥ 0} acting on B(H_{k}), k = 1, 2, . . .. For each n = 1, 2, … consider the type I factor

A_{n} = B(H_{1} ⊗ ⋯ ⊗ H_{n}),

and the semigroup of completely positive maps ϕ^{1} ⊗ ⋯ ⊗ ϕ^{n} defined on it by

(ϕ^{1} ⊗ ⋯ ⊗ ϕ^{n})_{t} = ϕ_{t}^{1} ⊗ ⋯ ⊗ ϕ_{t}^{n}, t ≥ 0.

One verifies easily that ϕ^{1} ⊗ ⋯ ⊗ ϕ^{n} is a CP semigroup. The natural embedding of A_{n} into A_{n+1} is given by the unital ∗-monomorphism A ∈ A_{n} ↦ A⊗1_{H_{n+1}} ∈ A_{n+1}, and since ϕ_{t}^{n+1}(1_{H_{n+1}}) = 1_{H_{n+1}} we have the following coherence

(2.1)  (ϕ^{1} ⊗ ⋯ ⊗ ϕ^{n+1})_{t}(A ⊗ 1_{H_{n+1}}) = (ϕ^{1} ⊗ ⋯ ⊗ ϕ^{n})_{t}(A) ⊗ 1_{H_{n+1}},

for every t ≥ 0, every A ∈ A_{n}, and every n = 1, 2, . . ..

Finally, we form the inductive limit of C^{∗}-algebras

A = lim_{⟶} A_{n}.

There is a natural way of identifying A_{n} with a subalgebra of A, and after making this identification one has A_{n} ⊆ A_{n+1} ⊆ A, and A is the norm closure of the unital ∗-subalgebra ∪_{n}A_{n}. Because of (2.1) there is a unique semigroup of unital completely positive maps ϕ = {ϕ_{t} : t ≥ 0} which acts as follows on A,

(2.2)  ϕ_{t} ↾_{A_{n}}= (ϕ^{1} ⊗ ⋯ ⊗ ϕ^{n})_{t},  t ≥ 0, n = 1, 2, ….

In order to discuss continuity in the time parameter of this semigroup, we require the notion of locally normal linear functionals on the C^{∗}-algebra A. A bounded linear functional ρ on A is said to be locally normal if its restriction to each subalgebra A_{n} is a normal linear functional on that type I factor. The set of all locally normal functionals is a norm-closed linear subspace of the dual space of A. Since every CP semigroup acting on a type I factor M is the adjoint of a strongly continuous semigroup acting on its predual M_{∗}, it follows that for every locally normal linear functional ρ on A, we have

(2.3)  lim_{t→t_{0}} ∥ρ ∘ ϕ_{t} ↾_{A_{n}} −ρ ∘ ϕ_{t_{0}} ↾_{A_{n}} ∥ = 0,  n = 1, 2, ….

We emphasize that one cannot expect t ↦ ρ ∘ ϕ_{t} to move continuously in the norm of the dual space of A, except in rather special circumstances. In any case, from (2.3) it follows that for every A ∈ A and every locally normal ρ, the function t ∈ [0, ∞) ↦ ρ(ϕ_{t}(A)) is continuous.

Suppose now that we choose a sequence ω_{k} of pure normal states ω_{k} of B(H_{k}), k = 1, 2, …, subject to the following hypothesis: there are positive constants A_{1}, A_{2}, … satisfying ∑_{k} A_{k} < ∞ and

(H)  ∥ω_{k} − ω_{k} ∘ ϕ_{t}^{k}∥ ≤ A_{k},  0 ≤ t ≤ 1, k = 1, 2, ….

For example, (H) will be satisfied when each space H_{k} is finite dimensional and the generators L_{k} of the semigroups ϕ^{k}, defined by ϕ_{t}^{k} = exp tL_{k}, t ≥ 0, satisfy ∑_{k} ∥ω_{k} ∘ L_{k}∥ < ∞ (see Corollary 1 below). Explicit examples will be discussed in section 3.

Since ω_{1} ⊗ ⋯ ⊗ ω_{n} is a normal pure state of A_{n} for every n, the product state ω̄ = ω_{1}⊗ω_{2}⊗… is a locally normal pure state of A. Applying the GNS construction to ω̄ we obtain a separable Hilbert space K, a faithful irreducible representation of A on K, and a distinguished unit vector Ω ∈ K. We identify A with its image in this representation, so that it becomes a unital strongly dense ∗-subalgebra A ⊆ B(K), and once this is done we have ω̄(A) = ⟨AΩ, Ω⟩ for A ∈ A. Following is the main result of this section.

Theorem A. Assume that hypothesis (H) is satisfied. Then for each t ≥ 0 the CP map ϕ_{t} defined on A by (2.2) extends uniquely to a normal completely positive linear map ϕ̃_{t} on B(K), and ϕ̃ = {ϕ̃_{t} : t ≥ 0} is a CP semigroup on B(K).

ϕ̃ is a pure CP semigroup on B(K) iff every ϕ_{k} is a pure CP semigroup on B(H_{k}), k = 1, 2, . . ..

Assuming that H_{k} is finite dimensional and ϕ^{k} has the normalized trace on B(H_{k}) as an absorbing state for every k = 1, 2, …, then ϕ̃ is pure and has no normal invariant state.

Before giving the proof, we require a more concrete realization of K as the infinite tensor product of the Hilbert spaces H_{1}, H_{2}, . . .. Our construction is similar to that in [vN] (see also [G]). For n = 1, 2, … let K_{n} be the n-fold tensor product of Hilbert spaces

K_{n} = H_{1} ⊗ H_{2} ⊗ ⋯ ⊗ H_{n}.

Choosing a unit vector f_{k} ∈ H_{k} such that ω_{k}(A) = ⟨Af_{k}, f_{k}⟩ for A ∈ B(H_{k}), we can map K_{n} isometrically into K_{n+1} as follows

ξ ∈ K_{n} ↦ ξ ⊗ f_{n+1} ∈ K_{n+1},

and this gives rise to an inductive system of Hilbert spaces. K is defined as the completion of the inductive limit of Hilbert spaces

K = \overline{lim_{⟶} K_{n}}.

Thus each space K_{n} is isometrically embedded in K and their union is dense. For a vector ξ ∈ K_{n} we denote its image in K with the suggestive notation

ξ ⊗ f_{n+1} ⊗ f_{n+2} ⊗ ⋯ = ξ ⊗ ⊗_{k=n+1}^{∞}f_{k}.

Note that parentheses come and go with impunity; for example, the equality

ξ ⊗ f_{n+1} ⊗ f_{n+2} ⊗ ⋯ = (ξ ⊗ f_{n+1}) ⊗ f_{n+2} ⊗ …

simply asserts that the image of ξ ∈ K_{n} in the limit space K is the same as the image of ξ ⊗ f_{n+1} ∈ K_{n+1} in the limit space K.

The C^{∗}-algebra A acts on K as follows. To define a representation of A it suffices to specify how an operator in A_{n} acts on K_{m} when m > n, and for an operator A ∈ A_{m} and a vector in K_{m} ⊆ K of the form η = ξ ⊗ f_{m+1} ⊗ f_{m+2} ⊗ … Aη is given by

Aη = [(A ⊗ 1_{H_{n+1}} ⊗ ⋯ ⊗ 1_{H_{m}})ξ] ⊗ f_{m+1} ⊗ f_{m+2} ⊗ ….

It is easily checked that this defines a representation of A on K, and that for the vector Ω ∈ K given by Ω = f_{1} ⊗ f_{2} ⊗ …, AΩ is dense in K and

(ω_{1} ⊗ ω_{2} ⊗ …)(A) = ⟨AΩ, Ω⟩ , A ∈ A.

Thus, we settle on this Hilbert space K, this “product vector” Ω ∈ K, and the realization of A ⊆ B(K) described in the preceding paragraphs.

This has the following consequence for the description of normal linear functionals on B(K). For any vector h ∈ K_{n} = H_{1} ⊗ H_{2} ⊗ ⋯ ⊗ H_{n} we may consider its image ξ = h⊗f_{n+1} ⊗f_{n+2} ⊗… in K , and the corresponding vector state on B(K)

ω_{ξ} = ω_{h} ⊗ ω_{n+1} ⊗ ω_{n+2} ⊗ ….

Since any normal linear functional ρ on A_{n} can be expressed as a series

ρ(A) = ∑_{k=1}^{∞} θ_{k} ⟨Au_{k}, v_{k}⟩

where u_{k}, v_{k} are unit vectors in H_{1} ⊗ ⋯ ⊗ H_{n} and (θ_{k}) is a summable sequence of nonnegative reals, a similar argument shows that there is a unique normal state on B(K) of the form

(2.4)  ρ̂ = ρ ⊗ ω_{n+1} ⊗ ω_{n+2} ⊗ ⋯ = ρ ⊗ ⊗_{k=n+1}^{∞}ω_{k},

The tensor product notation is intended merely to suggest the form this linear functional takes on when it is restricted to the dense C^{∗}-algebra A ⊆ B(K).

Lemma 2.1. For every normal linear functional (resp. state) σ of B(K) and every ϵ > 0, there is an n ∈ ℕ and a normal linear functional (resp. state) ρ on A_{n} such that

∥σ − ρ ⊗ ⊗_{k=n+1}^{∞}ω_{k}∥ < ϵ.

Proof. Let σ be a normal linear functional on B(K). We can approximate σ as closely as we wish with a finite linear combination of functionals of the form ω_{ξ,η}(T) = ⟨T ξ, η⟩ with ξ, η ∈ H. Thus it suffices to show that each ω_{ξ,η} can be so approximated. Fixing ξ and η in H, we can find an n ∈ ℕ and vectors ξ_{n}, η_{n} ∈ H_{n} such that both norms

∥ξ − ξ_{n} ⊗ f_{n+1} ⊗ f_{n+2} ⊗ … ∥, ∥η − η_{n} ⊗ f_{n+1} ⊗ f_{n+2} ⊗ … ∥

are less than ϵ/2. It follows that

∥ω_{ξ,η} − ω_{ξ_{n},η_{n}} ⊗ ⊗_{k=n+1}^{∞}ω_{k}∥ < ϵ,

as asserted.

The argument for normal states is similar, after one notes that every normal state σ of B(K) can be expressed as a convex combination of vector states

σ = θ_{1}ω_{ξ_{1},ξ_{1}} + θ_{2}ω_{ξ_{2},ξ_{2}} + …

where θ_{k} ≥ 0 and ∑_{k} θ_{k} = 1. □

Since A ⊆ B(K) it makes sense to speak of normal states and normal linear functionals on A; for example, a normal state of A is the restriction to A of a normal state of B(K). We remark that a straightforward application of Kaplansky’s density theorem shows that the restriction map ρ ↦ ρ ↾_{A} is an order-preserving isometric isomorphism of the Banach space of all normal linear functionals of B(K) onto the normal subspace of the dual of A. In particular the normal subspace of A′ is closed in the norm of A′.

Lemma 2.2. Let θ : A → A be a unital completely positive map with the property that ρ ∘ θ on A extends to a normal state on B(K) for every normal state ρ on B(K). Then θ extends uniquely to a normal CP map θ̃ on B(K).

proof. The uniqueness of the extension is clear from the fact that A is weak^{∗}-dense in B(K). Morever, if there is a normal linear map θ̃ which extends θ then θ̃ must be completely positive. Thus we need only show that θ can be extended to a normal map on B(K).

Let L^{1}(K) denote the Banach space of all trace class operators on B(K) and choose T ∈ L^{1}(K). Then ρ(B) = trace(T B) defines a normal linear functional on B(K). By hypothesis A ∈ A ↦ ρ(θ(A)) can be extended to a normal linear functional on B(K), hence there is a unique trace class operator ψ(T) ∈ L^{1}(K) such that

trace(ψ(T)A) = trace(T θ(A)), A ∈ A.

ψ is obviously a linear map of L^{1}(K) into itself. Since the unit ball of A is strongly dense in the unit ball of B(K) and ∥θ∥ ≤ 1, the formula itself implies that

trace |ψ(T)| = sup_{∥A∥≤1} |trace(ψ(T)A)| = sup_{∥A∥≤1} |trace(T θ(A))| ≤ trace |T |

for every T ∈ L^{1}(K), hence ∥ψ∥ ≤ 1. The adjoint ψ′ : B(K) → B(K) defines a normal linear map on B(K) which satisfies trace(T ψ′(A)) = trace(T θ(A)) for A ∈ A and all T ∈ L^{1}(K), thus ψ′ extends θ. □

proof of Theorem A. Fix t ≥ 0. We show first that ϕ_{t} : A → A can be extended (necessarily uniquely) to a normal completely positive map on B(K). Note that it suffices to prove this for 0 ≤ t ≤ 1, because the semigroup property implies that ϕ_{nt} = (ϕ_{t})^{n} for every n = 1, 2, … and a composition of normal maps on B(K) is a normal map. Thus we assume that t ∈ [0, 1].

By Lemma 2.2, it is enough to show that for every normal linear functional ρ on B(K), ρ ∘ θ_{t} is a normal functional on A. Lemma 2.1 implies that ρ can be approximated in norm with a decomposable linear functional of the form

ρ_{0} = ψ ⊗ ⊗_{k=n+1}^{∞}ω_{k}

for some n ∈ ℕ, where ψ is a normal linear functional on A_{n} satifying ∥ψ∥ = ∥ρ_{0}∥. Notice that by adjusting ψ appropriately, we may also choose n as large as we wish in this representation for ρ_{0}. Since the normal part of the dual of A is norm-closed, it suffices to show that ρ_{0} ∘ ϕ_{t} belongs to the normal part of A′.

For such a decomposable ρ_{0}, we have

ρ_{0} ∘ ϕ_{t} = [ψ ∘ (ϕ_{t}^{1} ⊗ ⋯ ⊗ ϕ_{t}^{n})] ⊗ ⊗_{k=n+1}^{∞}ω_{k} ∘ ϕ_{t}^{k}.

Considering the normal linear functional on B(K)

ν = [ψ ∘ (ϕ_{t}^{1} ⊗ ⋯ ⊗ ϕ_{t}^{n})] ⊗ ⊗_{k=n+1}^{∞}ω_{k}

and the fact that ∥ψ ∘ (ϕ_{t}^{1} ⊗ ⋯ ⊗ ϕ_{t}^{n})∥ ≤ ∥ψ∥ = ∥ρ_{0}∥, we have

∥ρ_{0} ∘ ϕ_{t} − ν∥ ≤ ∥ρ_{0}∥ · ∥( ⊗_{k=n+1}^{∞}ω_{k} ∘ ϕ_{t}^{k} − ⊗_{k=n+1}^{∞}ω_{k}) ↾_{A} ∥.

Since

∥( ⊗_{k=n+1}^{∞}ω_{k} ∘ ϕ_{t}^{k} − ⊗_{k=n+1}^{∞}ω_{k}) ↾_{A} ∥ ≤ ∑_{k=n+1}^{∞} ∥ω_{k} ∘ ϕ_{t}^{k} − ω_{k}∥

and since for 0 ≤ t ≤ 1 the series on the right is the tail of a convergent series by (H), it follows that ∥ρ_{0} −ν∥ can be made arbitrarily small by choosing n sufficiently large. Since ν is normal, ρ_{0} ∘ ϕ_{t} must be normal, as required.

Thus we may extend each ϕ_{t} uniquely to a normal completely positive map ϕ̃_{t} of B(K). We prove next that the extended semigroup is continuous in the time parameter in the sense that for every normal linear functional ρ on B(K)

lim_{t→0} ∥ρ ∘ ϕ̃_{t} − ρ∥ = 0.

Since we can approximate ρ arbitrarily closely in norm with decomposable functionals of the form

ρ_{0} = ψ ⊗ ⊗_{k=n+1}^{∞}ω_{k}

where n ∈ ℕ and ψ ∈ A′_{n} and since ∥ϕ̃_{t}∥ ≤ 1 for every t, it suffices to show that ∥ρ_{0} ∘ ϕ̃_{t} − ρ_{0}∥ → 0 as t → 0, for ρ_{0} of this form.

As in the preceding argument, we may suppose that n is as large as we please; moreover the norm of any normal linear functional on B(K) agrees with the norm of its restriction to A. Writing

ρ_{0} ∘ ϕ̃_{t} − ρ_{0} = (ρ_{0} ∘ ϕ̃_{t} − ψ ⊗ ⊗_{k=n+1}^{∞}ω_{k} ∘ ϕ_{t}^{k}) + (ψ ⊗ ⊗_{k=n+1}^{∞}ω_{k} ∘ ϕ_{t}^{k} − ρ_{0}),

we estimate the norm of (ρ_{0} ∘ ϕ̃_{t} − ρ_{0}) ↾_{A} as follows

∥ρ_{0} ∘ ϕ̃_{t} − ρ_{0}∥ ≤∥ψ ∘ (ϕ_{t}^{1} ⊗ ⋯ ⊗ ϕ_{t}^{n}) − ψ∥ · ∥ ⊗_{k=n+1}^{∞} ω_{k} ∘ ϕ_{t}^{k}∥

+ ∥ψ∥ · ∥ ⊗_{k=n+1}^{∞} ω_{k} ∘ ϕ_{t}^{k} − ⊗_{k=n+1}^{∞}ω_{k}∥

≤∥ψ ∘ (ϕ_{t}^{1} ⊗ ⋯ ⊗ ϕ_{t}^{n}) − ψ∥ + ∥ ⊗_{k=n+1}^{∞} ω_{k} ∘ ϕ_{t}^{k} − ⊗_{k=n+1}^{∞}ω_{k}∥

The second term on the right can be estimated as in the preceding argument,

∥ ⊗_{k=n+1}^{∞} ω_{k} ∘ ϕ_{t}^{k} − ⊗_{k=n+1}^{∞}ω_{k}∥ ≤ ∑_{k=n+1}^{∞} ∥ω_{k} ∘ ϕ_{t}^{k} − ω_{k}∥ ≤ ∑_{k=n+1}^{∞} A_{k},

because of (H). Since ϕ^{1} ⊗ ⋯ ⊗ ϕ^{n} is a CP semigroup acting on B(H_{1} ⊗ ⋯ ⊗ H_{n}), we have

lim sup_{t→0} ∥ρ_{0} ∘ ϕ̃_{t} − ρ_{0}∥ ≤ lim sup_{t→0} ∥ψ ∘ (ϕ_{t}^{1} ⊗ ⋯ ⊗ ϕ_{t}^{n}) − ψ∥ + ∑_{k=n+1}^{∞} A_{k} = ∑_{k=n+1}^{∞} A_{k},

and since we are free to choose n as large as we please, the right side can be made as small as we please because ∑_{k} A_{k} converges. Thus ϕ̃ is a CP semigroup.

Assuming now that each ϕ^{k} is a pure CP semigroup, we prove next that ϕ̃ is pure. That result requires that we first discuss the behavior of pure CP semigroups under finite tensor products. For a Hilbert space H, we write L_{0}^{1}(H) for the space of all trace class operators T ∈ L^{1}(H) satisfying trace T = 0.

Remark. The significance of the space L_{0}^{1}(H) for pure CP semigroups acting on B(H) is as follows. Notice first that a trace class operator T on H belongs to L_{0}^{1}(H) iff its associated linear functional σ(A) = trace(T A), A ∈ B(H), satisfies σ(1) = 0. On the other hand, a normal linear functional σ satisfies σ(1) = 0 iff it admits a decomposition

σ = λ · (ρ_{1} − ρ_{2}) + μ · \sqrt{−1} · (ρ_{3} − ρ_{4}),

where λ, μ are real scalars and each ρ_{k} is a normal state. We conclude: a CP semigroup ϕ = {ϕ_{t} : t ≥ 0} acting on B(H) is pure iff for every normal linear functional σ satisfying σ(1) = 0, we have

(2.5)  lim_{t→∞} ∥σ ∘ ϕ_{t}∥ = 0.

Lemma 2.3. For Hilbert spaces H_{1} and H_{2},

L_{0}^{1}(H_{1} ⊗ H_{2}) = \overline{span}\{T_{1} ⊗ S_{2} + S_{1} ⊗ T_{2} : T_{k} ∈ L^{1}(H_{k}), S_{k} ∈ L_{0}^{1}(H_{k})\}.

proof. The inclusion ⊇ is obvious. For the opposite inclusion, let A ∈ B(H_{1} ⊗ H_{2}) be a bounded operator satisfying trace(RA) = 0 for every trace class operator R of the form R = T_{1} ⊗ S_{2} or R = S_{1} ⊗ T_{2} where S_{k} ∈ L_{0}^{1}(H_{k}) and T_{k} ∈ L^{1}(H_{k}), k = 1, 2. Considering the natural duality between bounded operators and trace class operators, it is enough to show that A is a scalar multiple of the identity.

Indeed, we claim that from the hypothesis

trace(A(S_{1} ⊗ T_{2})) = 0, for all S_{1} ∈ L_{0}^{1}(H_{1}), T_{2} ∈ L^{1}(H_{2})

it follows that A ∈ (B(H_{1}) ⊗ 1_{H_{2}})′. To see that, choose X ∈ B(H_{1}) and consider the commutator [X ⊗ 1_{H_{2}}, A] = (X ⊗ 1_{H_{2}})A − A(X ⊗ 1_{H_{2}}). Then for any pair of trace class operators T_{k} ∈ L^{1}(H_{k}), k = 1, 2 we have

trace([X ⊗ 1_{H_{2}}, A]T_{1} ⊗ T_{2}) = trace(A(T_{1}X − XT_{1}) ⊗ T_{2}) = 0,

because the commutator T_{1}X − XT_{1} belongs to L_{0}^{1}(H_{1}). Since L^{1}(H_{1} ⊗ H_{2}) is the closed linear span of {T_{1} ⊗ T_{2} : T_{k} ∈ L^{1}(H_{k})}, [X ⊗ 1_{H_{2}}, A] = 0 follows.

Similarly, the hypothesis

trace(A(T_{1} ⊗ S_{2})) = 0, for all T_{1} ∈ L^{1}(H_{1}), S_{2} ∈ L_{0}^{1}(H_{2})

implies that A ∈ (1_{H_{1}} ⊗ B(H_{2}))′. Hence

A ∈ (B(H_{1}) ⊗ 1_{H_{2}})′ ∩ (1_{H_{1}} ⊗ B(H_{2}))′ = (B(H_{1}) ⊗ B(H_{2}))′ = ℂ · 1_{H_{1}⊗H_{2}},

as required. □

Lemma 2.4. Let ϕ^{1} and ϕ^{2} be pure CP semigroups acting on B(H_{1}) and B(H_{2}) respectively. Then the tensor product ϕ^{1} ⊗ ϕ^{2} is a pure CP semigroup acting on B(H_{1} ⊗ H_{2}).

proof. We deduce this from Lemma 2.3 as follows. According to the characterization (2.5), it is enough to show that for every normal linear functional σ on B(H_{1} ⊗ H_{2}) satisfying σ(1) = 0, we have ∥σ ∘ (ϕ_{t}^{1} ⊗ ϕ_{t}^{2})∥ → 0 as t → ∞. Because of Lemma 2.3, σ can be closely approximated in norm with a finite linear combination of functionals of the form ρ_{1} ⊗ρ_{2} where ρ_{k} is a normal linear functional on B(H_{k}) and where either ρ_{1} or ρ_{2} annihilates the identity operator on its respective space.

But for a functional of the form σ = ρ_{1} ⊗ ρ_{2} we have

∥(ρ_{1} ⊗ ρ_{2}) ∘ (ϕ_{t}^{1} ⊗ ϕ_{t}^{2})∥ = ∥ρ_{1} ∘ ϕ_{t}^{1}∥ · ∥ρ_{2} ∘ ϕ_{t}^{2}∥,

so that if either ρ_{1}(1_{H_{1}}) = 0 or ρ_{2}(1_{H_{2}}) = 0, then because both ϕ^{1} and ϕ^{2} are pure we conclude from (2.5) that

lim_{t→∞} ∥ρ_{1} ∘ ϕ_{t}^{1}∥ · ∥ρ_{2} ∘ ϕ_{t}^{2}∥ = 0.

After taking finite linear combinations of such functionals ρ_{1} ⊗ ρ_{2} and then passing to norm limits, it follows that ∥σ ∘ (ϕ_{t}^{1} ⊗ ϕ_{t}^{2})∥ → 0 as t → ∞ for every normal linear functional σ satisfying σ(1) = 0. □

Returning now to the proof of Theorem A, suppose that each of the given sequence of CP semigroups ϕ^{k} is pure. ¿From Lemma 2.4 it follows that every finite tensor product ϕ^{1} ⊗⋯⊗ϕ^{n} is a pure CP semigroup acting on A_{n}, n = 1, 2, . . .. We now show that the infinite tensor product ϕ̃ is pure. For that, fix a pair of normal states ρ, σ on B(K). We have to show that

lim_{t→∞} ∥ρ ∘ ϕ̃_{t} − σ ∘ ϕ̃_{t}∥ = 0.

Again, we approximate ρ and σ closely in norm with normal product states ρ_{0} and σ_{0} respectively

ρ_{0} = ρ^{n} ⊗ ⊗_{k=n+1}^{∞}ω_{k}, σ_{0} = σ^{n} ⊗ ⊗_{k=n+1}^{∞}ω_{k}

where ρ^{n} and σ^{n} are states of A_{n} and n is a positive integer, as large as we wish. Estimating as above, we have

∥(ρ_{0} − σ_{0}) ∘ ϕ̃_{t} ↾_{A} ∥ ≤ ∥(ρ^{n} − σ^{n}) ∘ (ϕ_{t}^{1} ⊗ ⋯ ⊗ ϕ_{t}^{n}) ↾_{A_{n}} ∥ · ∥ ⊗_{k=n+1}^{∞} ω_{k} ∘ ϕ_{t}^{k}∥
≤ ∥(ρ^{n} − σ^{n}) ∘ (ϕ_{t}^{1} ⊗ ⋯ ⊗ ϕ_{t}^{n}) ↾_{A_{n}} ∥

and the right side tends to zero as t → ∞ because ϕ^{1} ⊗ ⋯ ⊗ ϕ^{n} is a pure CP semigroup on A_{n}.

The converse assertion, namely that purity of ϕ̃ on B(K) implies that each ϕ^{k} is a pure CP semigroup on B(H_{k}) is straightforward, and we omit the proof. □

When the Hilbert spaces H_{k} are all finite dimensional, there is a more concrete criterion in terms of the generators of the individual semigroups ϕ^{k} on B(H_{k}).

Corollary 1. For every k = 1, 2, … let ϕ^{k} be a CP semigroup acting on a full matrix algebra M_{k} which has the normalized trace on M_{k} as an absorbing state, and let ω_{k} be a pure state of M_{k}. Let ω̄ = ω_{1} ⊗ ω_{2} ⊗ … be the asociated product state on the UHF algebra A = M_{1} ⊗ M_{2} ⊗ … and let ϕ = {ϕ_{t} : t ≥ 0} be the product semigroup, acting on A by ϕ_{t} = ϕ_{t}^{1} ⊗ ϕ_{t}^{2} ⊗ …, t ≥ 0.

Let L_{k} : M_{k} → M_{k} be the generator of ϕ^{k}, ϕ_{t}^{k} = exp tL_{k}, t ≥ 0, and assume that

(2.6)  ∑_{k=1}^{∞} ∥ω_{k} ∘ L_{k}∥ < ∞.

The GNS construction applied to ω̄ realizes A as an irreducible C^{∗}-algebra acting on a Hilbert space K with the property that ϕ extends uniquely to a normal CP semigroup ϕ̃ acting on B(K) which is pure and has no normal invariant states.

Remarks. Regarding the summability hypothesis (2.6), consider any CP semigroup ϕ = {ϕ_{t} : t ≥ 0} acting on a matrix algebra M and let L be the generator of ϕ, ϕ_{t} = exp tL, t ≥ 0. A state ω is invariant in the sense that ω ∘ ϕ_{t} = ω for all t ≥ 0 iff ω ∘ L = 0. Thus one may regard ω as being “approximately” invariant when ∥ω ∘ L∥ is small. In the setting of Corollary 1, the tracial state of M_{k} is the unique ϕ^{k}-invariant state of M_{k}, thus no pure state can be invariant. However, the hypothesis (2.6) means that the sequence of pure states ω_{k} should be chosen so that they are “approximately” invariant in the sense that the sum (2.6) converges. Notice however that (2.6) will be satisfied with an arbitrary sequence of pure states ω_{1}, ω_{2}, … whenever ∥L_{k}∥ tends to zero fast enough that ∑_{k} ∥L_{k}∥ converges.

proof of Corollary 1. We show first that ϕ̃ extends uniquely to a pure CP semigroup on B(K). In view of Theorem A it suffices to show that the hypothesis (2.6) implies that (H) is satisfied. For that, we claim that for any CP semigroup acting on a matrix algebra M with generator L, one has

(2.7)  sup_{0≤t≤1} ∥ω ∘ ϕ_{t} − ω∥ ≤ ∥ω ∘ L∥,

for every state ω of M. Indeed, for fixed A ∈ M satisfying ∥A∥ ≤ 1,

ω ∘ ϕ_{t}(A) − ω(A) = ∫_{0}^{t} \frac{d}{ds}ω ∘ ϕ_{s}(A) , ds = ω ∘ L(∫_{0}^{t} ϕ_{s}(A) ds).

Since

∥ ∫_{0}^{t} ϕ_{s}(A) ds∥ ≤ ∫_{0}^{t} ∥ϕ_{s}(A)∥ ds ≤ t · ∥A∥ ≤ t,

we have |ω ∘ ϕ_{t}(A) − ω(A)| ≤ ∥ω ∘ L∥ for every t ∈ [0, 1], and (2.7) follows.

It remains only to show that ϕ̃ has no normal invariant state, and for that we claim that the normalized trace τ on the UHF algebra A = M_{1} ⊗ M_{2} ⊗ … is the unique state of A which is invariant under the action of the product semigroup ϕ^{1} ⊗ ϕ^{2} … acting on A. Indeed, Lemma 2.3 implies that for every n = 1, 2, …, the restriction of ϕ^{1} ⊗ ϕ^{2} ⊗ … to A_{n} = B(H_{1} ⊗ ⋯ ⊗ H_{n}) is pure, and since by hypothesis each ϕ^{k} leaves the trace of B(H_{k}) invariant, it follows that the trace of B(H_{1} ⊗ ⋯ ⊗ H_{n}) is invariant under the action of this restricted CP semigroup, hence the trace on B(H_{1} ⊗ ⋯ ⊗ H_{n}) is an absorbing state. In particular it is the unique invariant state. It follows that τ is the unique invariant state of A.

Now if ρ is a ϕ̃-invariant normal state of B(K) then the restriction of ρ to A is an invariant state of A which, by the preceding paragraph, must be the normalized trace τ of A. But τ cannot belong to the normal part of the dual of A in any irreducible representation of A, contradicting the existence of ρ. □

Corollary 2. Let M = M_{N}(ℂ) be a matrix algebra and let θ = {θ_{t} : t ≥ 0} be a CP semigroup on M which has the normalized trace as an absorbing state. Let λ_{1}, λ_{2}, … be a sequence of positive numbers satisfying ∑_{n} λ_{n} < ∞. Consider the UHF algebra A = M ⊗ M ⊗ … and the CP semigroup acting on it as follows

ϕ_{t} = θ_{λ_{1}t} ⊗ θ_{λ_{2}t} ⊗ ….

Then for every sequence of pure state ω_{k} of M the product state ω_{1} ⊗ ω_{2} ⊗ … gives rise to an irreducible representation of A on a Hilbert space H with the property that ϕ extends uniquely to a CP semigroup acting on B(H) which has no normal invariant states.

proof. Letting L_{k} be the generator of ϕ_{t}^{k} = θ_{λ_{k}t}, one sees that L_{k} is related to the generator M of θ by L_{k} = λ_{k}M, and hence

∑_{k=1}^{∞} ∥ω_{k} ∘ L_{k}∥ ≤ ∑_{k=1}^{∞} ∥L_{k}∥ = ∥M∥ · ∑_{k=1}^{∞} λ_{k} < ∞,

and the conclusion is immediate from Corollary 1. □

3. Examples.

In order to apply the results of the preceding section one must start with a sequence of CP semigroups acting on matrix algebras, each of which is pure and has the normalized trace as an absorbing state. In this section we give concrete examples of the latter and discuss how such examples can be constructed in general.

Remarks. We have already pointed out that if a CP semigroup ϕ = {ϕ_{t} : t ≥ 0} acting on B(H) has an absorbing state, then it must be pure. Significantly, in the simplest cases in which H is finite dimensional, the converse is true as well. Indeed, any pure CP semigroup acting on a matrix algebra M must have an invariant state ω, by the Markov-Kakutani fixed point theorem. It follows that for any state ρ on M we have

lim_{t→∞} ∥ρ ∘ ϕ_{t} − ω∥ = lim_{t→∞} ∥ρ ∘ ϕ_{t} − ω ∘ ϕ_{t}∥ = 0,

because ϕ is pure. Hence ω must be an absorbing state.

We begin with a concrete example and then put that into a more general context.

Example 3.1. In [Bi] P. Biane defines a CP semigroup B = {B_{t} : t ≥ 0}, called an Ornstein-Uhlenbeck semigroup, on finite-dimensional C^{∗}-algebras A generated by finitely many hermitian unitary elements u_{1}, u_{2}, …, u_{n}, n ∈ ℕ, which pairwise either commute or anticommute. The action of the semigroup on ordered words w = u_{k_{1}}u_{k_{2}} … u_{k_{j}}, k_{1} < k_{2} < ⋯ < k_{j} in A is as follows, B_{t}(w) = exp(−tj)w. Assuming that n = 2N is even and the u_{k} anticommute, their generated C^{∗}-algebra A is isomorphic to the matrix algebra M_{2^{N}} (ℂ); and since for every nontrivial word w we have

lim_{t→∞} B_{t}(w) = 0,

it follows that for every A ∈ A we have

lim_{t→∞} B_{t}(A) = τ(A)1.

Thus, B is a pure CP semigroup acting on A which has the normalized trace as an absorbing state.

The following results show how to construct many such examples.

Proposition 3.1. Let M be a matrix algebra and let Q be a unit-preserving completely positive map of M into itself which preserves the trace in the sense that τ ∘ Q = τ and is ergodic in the sense that

(3.1)  Q(A) = A ⟹ A ∈ ℂ · 1.

Then L(A) = Q(A) − A is the generator of a pure CP semigroup which has the normalized trace as an absorbing state.

Proof. Let ϕ_{t} = exp tL where L(A) = Q(A) − A. Since ϕ_{t} = e^{−t} exp tQ, it is clear that ϕ is a CP semigroup, and it leaves the trace invariant because τ ∘ L = 0. Condition (3.1) implies that the only operators A satisfying ϕ_{t}(A) = A for all t are scalars, i.e., ϕ acts ergodically on M. By Theorem 4.4 of [A2] ϕ is pure, and thus τ must be an absorbing state. □

Remark. It is well-known [EL] that the most general unit-preserving completely positive linear map on a matrix algebra M may be constructed as follows

Q(A) = V_{1}AV_{1}^{∗} + ⋯ + V_{r}AV_{r}^{∗},

where V_{1}, …, V_{r} is an arbitrary sequence of elements of M satisfying V_{1}V_{1}^{∗} + ⋯ + V_{r}V_{r}^{∗} = 1. For such a map Q the ergodicity requirement (3.1) is satisfied iff the set {V_{k}} is irreducible in the sense that

{V_{1}, V_{1}^{∗}, …, V_{r}, V_{r}^{∗}}′ = ℂ1.

This follows, for example, from Theorem 4.4 of [A2].

Remark. As the simplest nontrivial example of the semigroups of Example 3.1 above, consider the semigroup B = {B_{t} : t ≥ 0} acting on the C^{∗}-algebra generated by a pair of anticommuting self-adjoint unitaries u_{1}, u_{2}. Up to conjugacy one may take the C^{∗}-algebra to be M_{2}(ℂ), and for the generator of the semigroup {B_{t} : t ≥ 0} one can take the map L(A) = Q(A) − A where Q is the composition θ ∘ E of the natural conditional expectation E of M_{2}(ℂ) onto the diagonal subalgebra and θ is the order 2 automorphism θ(A) = RAR ,

R = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) .

We omit these computations, but one can check quite easily that the map Q = θ∘E satisfies the hypotheses of Proposition 3.1.

4. Applications to E_{0}-semigroups.

In this section we show how the CP semigroups constructed in section 2 can be dilated so as to give examples of pure E_{0}-semigroups which have no normal invariant states.

It is important to distinguish between the dilation theory that is associated with CP semigroups and the simpler dilation theory associated with Stinespring’s theorem. In order to discuss the issue, we take the simplest case in which ϕ is a normal completely positive map acting on B(H) satisfying ϕ(1) = 1. Stinespring’s theorem implies that there is a Hilbert space H̃ containing H and a representation π : B(H) → B(H̃) satisfying π(1_{H}) = 1H̃, such that for P = P_{H} we have

ϕ(A) = P π(A) ↾_{H}, A ∈ B(H).

Since ϕ is normal, one can arrange that π is a normal representation. We abuse notation slightly by writing ϕ(A) = P π(A)P, after identifying B(H) with the corner P B(H̃)P of B(H̃).

Useful as the Stinespring representation is, it is inadequate for our purposes. Instead, we require a Hilbert space H̃ ⊇ H and a normal ∗-endomorphism α of B(H̃) into itself, such that α(1) = 1, α(P) ≥ P for P = P_{H} and

(4.1)  ϕ(A) = P α(A)P,  A ∈ B(H) = P B(H̃)P.

One verifies readily that the condition α(P) ≥ P implies that (4.1) is valid for all powers in the sense that

(4.2)  ϕ^{n}(A) = P α^{n}(A)P,  n = 0, 1, 2, ….

Such a representation of ϕ cannot be deduced from Stinespring’s theorem. It can be deduced from a general dilation theorem for (finite or infinite) sequences of (perhaps noncommuting) operators V_{1}, V_{2}, ⋯ ∈ B(H) which satisfy

V_{1}V_{1}^{∗} + V_{2}V_{2}^{∗} + ⋯ = 1_{H}.

That dilation theorem implies that there is a sequence of isometries U_{1}, U_{2}, … on a larger Hilbert space H̃ ⊇ H satisfying U_{1}U_{1}^{∗} + U_{2}U_{2}^{∗} + ⋯ = 1H̃, with the property that U_{k}^{∗}H ⊆ H and V_{k} is the compression of U_{k} to H, for every k. The relation between all of these operators, ϕ, and α is

ϕ(A) = V_{1}AV_{1}^{∗} + V_{2}AV_{2}^{∗} + …, A ∈ B(H),
α(B) = U_{1}BU_{1}^{∗} + U_{2}BU_{2}^{∗} + … B ∈ B(H̃).

We remark that the existence of operator dilations of this type originated in papers of Frazho [Fr] and Bunce [Bu], and have been extensively studied.

In [B1] (also see [B2], [B3], [BF]) B. V. R. Bhat proved a general result which implies that every CP semigroup acting on B(H) can be dilated to an E_{0}-semigroup acting on B(H̃) in a sense parallel to (4.2) above. There is a notion of “minimal” dilation, and a minimal dilation of a CP semigroup is unique up to conjugacy. The details are as follows.

We follow the approach of [A5]. Let H̃ be a Hilbert space, and let α = {α_{t} : t ≥ 0} be an E_{0}-semigroup acting on B(H̃). A projection P ∈ B(H̃) is said to be increasing if P ≤ α_{t}(P) for all t ≥ 0. Assuming that P is increasing, it is not hard to show that the family of completely positive maps ϕ = {ϕ_{t} : t ≥ 0} defined on P B(H̃)P = B(P H̃) by

ϕ_{t}(x) = P α_{t}(a)P, t ≥ 0

satisfies the semigroup property, thus it defines a CP semigroup on B(P H̃). ϕ is called a compression of α and α is called a dilation of ϕ. α is said to be a minimal dilation of ϕ if the identity is the only increasing projection Q ∈ B(H̃), Q ≥ P, for which the compression of α to QB(H̃)Q is an E_{0}-semigroup. In this context, Bhat’s theorem has the following consequence (see [A5] for this formulation).

Theorem 4.1. Let ϕ be a CP semigroup on B(H). There exists a dilation α of ϕ on B(H̃), for some Hilbert space H̃ containing H, such that α is a minimal dilation of ϕ. Any two minimal dilations of ϕ are conjugate.

Thus we may speak of the minimal dilation of a CP semigroup. As shown in Proposition 3.5 of [A2], a minimal dilation α of a pure CP semigroup ϕ inherits from ϕ the property of being pure. If ϕ, in addition to being pure, has no invariant normal states, then α has no invariant normal states. Combining these results with Theorem A of Section 2 yields the following.

Theorem 4.2. Let α be the minimal dilation of a CP semigroup ϕ = {ϕ_{t} : t ≥ 0} on B(H) which arises from an infinite tensor product construction satisfying the hypotheses of Theorem A. Then α is pure and has no invariant normal states.

We conclude this section by showing that the minimal dilation α of ϕ has a realization as an infinite tensor product of E_{0}-semigroups (hence of CP semigroups) in the sense of section 2. We show also that in certain situations α is completely spatial. This result depends upon the fact, [A6], [P1] that the minimal dilation of a CP semigroup on a matrix algebra is completely spatial.

Let (ϕ^{k}) be a sequence of CP semigroups on matrix algebras M_{k} satisfying (H). We view M_{k} as B(H_{k}) for a finite dimensional Hilbert space H_{k}. For each k ∈ ℕ let α^{k}, acting on the algebra B(H̃_{k}), be a minimal dilation of ϕ^{k}.

First we observe that the pure state ω_{k} = ⟨ · f_{k}, f_{k}⟩ on B(H_{k}) extends in an obvious way to a pure state (which we’ll also call ω_{k}) on B(H̃_{k}) because H̃_{k} ⊇ H_{k}, and if the ϕ^{k} satisfy (H) then so do the α^{k}. For fixed k, let p be the projection of H̃_{k} onto H_{k}. We have

∥ω_{k} − ω_{k} ∘ α_{t}^{k}∥ = sup_{x∈B(H̃_{k})_{1}}|ω_{k}(x) − ⟨α_{t}^{k}(x)f_{k}, f_{k}⟩|.

Since p is increasing for α^{k} it follows that ϕ_{t}^{k}(pxp) = pα_{t}^{k}(pxp)p = pα_{t}^{k}(x)p, and since f_{k} = pf_{k}, the right side of the preceding formula is

sup_{x∈B(H̃_{k})_{1}}|ω_{k}(x) − ⟨ϕ_{t}^{(k)}(pxp)f_{k}, f_{k}⟩| ≤ ∥(ω_{k} − ω_{k} ∘ ϕ_{t}^{(k)}) ↾_{M_{k}} ∥∥x∥ ≤ A_{k},

where A_{k} is the constant in (H) which dominates ∥(ω_{k}−ω_{k}∘ϕ_{t}^{k}) ↾_{M_{k}} ∥ for 0 ≤ t ≤ 1. Continuing the argument along the lines of section 2, we conclude that the infinite tensor product ⊗_{k=1}^{∞}α_{k} exists as a CP semigroup. It is obviously an E_{0}-semigroup

which serves as a dilation of the CP semigroup ϕ = ⊗_{k=1}^{∞}ϕ_{k}. In fact, we have the

following.

Theorem 4.3. The dilation α on B(H̃) is a minimal dilation of ϕ.

Remark. It is straightforward to show, by using the criteria for minimality obtained in Theorem B of [A5], that the finite tensor product of minimal dilations of CP semigroups is itself a minimal dilation. The proof below relies on this result to establish the case for infinite tensor products.

proof. Let P ∈ B(H̃) be the projection onto the subspace H on which ϕ acts. Then P compresses α down to ϕ. Let P_{+} be the orthogonal projection in H̃ onto the closed linear span H_{+} of the vectors of the form

(4.3)  α_{t_{1}}(a_{1})α_{t_{2}}(a_{2}) ⋯ α_{t_{n}}(a_{n})ξ,

where ξ runs over H and the a_{j}’s lie in B(H). By [A5] P_{+} compresses α to a minimal dilation of ϕ. To show that α is minimal, then, it suffices to show that P_{+} is the identity in B(H̃), and to establish this it suffices to show that all product vectors of the form ξ_{1} ⊗ξ_{2} ⊗⋯⊗ξ_{m} ⊗f_{m+1} ⊗f_{m+2} ⊗⋯ are in the range of P_{+} for all vectors ξ_{k} ∈ H̃_{k}. Set ξ = ⊗_{k=1}^{∞}f_{k}. For fixed m ∈ ℕ consider the linear span of all terms of

the form (4.3) where a_{j} ∈ ( ⊗_{k=1}^{m}M_{k}) ⊗ (I ⊗ I ⊗ ⋯) for all j. By the remark above

⊗_{k=1}^{m}α^{(k)} on ⊗_{k=1}^{m}B(H̃_{k}) is a minimal dilation of ⊗_{k=1}^{m}ϕ^{(k)} on ⊗_{k=1}^{m}B(H_{k}), and therefore

ξ_{1} ⊗ ξ_{2} ⊗ ⋯ ⊗ ξ_{m} ⊗ f_{m+1} ⊗ f_{m+2} ⊗ ⋯ is in the range of P_{+}. But then, since m is arbitrary, all of H̃ is in range of P_{+}, so p_{+} = 1 and α is a minimal dilation of ϕ. □

We conjecture that the minimal dilations of CP semigroups constructed as in section 2 are all are all conjugate to a cocycle perturbation of a CAR/CCR flow. The following confirms the conjecture for a large family of CP semigroups satisfying the hypotheses of Corollary 2.

Theorem 4.4. Given a sequence of positive numbers λ_{1}, λ_{2}, …, let ϕ be a CP semigroup acting on a matrix algebra M. Let ϕ̃ be the infinite tensor product

ϕ̃_{t} = ϕ_{λ_{1}t} ⊗ ϕ_{λ_{2}t} ⊗ …

acting on the UHF algebra A = M ⊗ M ⊗ . . .. Let ω = ω_{1} ⊗ ω_{2} ⊗ ⋯ be a pure product state on A which induces an irreducible representation of A on B(H). If ∑_{n} λ_{n}^{1/2} < ∞, then the minimal dilation α̃ of the unique extension of ϕ̃ to B(H) is a cocycle perturbation of a CCR/CAR flow.

Remark. In [A1] one of us characterized the CAR/CCR flows as being those E_{0}- semigroups which are completely spatial (see the following definition). Our proof of the theorem rests on the verification that E_{0}-semigroups satisfying the hypotheses of the theorem are completely spatial. First we recall the following relevant definitions from [A1].

Definition 4.1. A strongly continuous one-parameter semigroup U = {U_{t} : t ≥ 0} of operators in B(H̃) is called a unit of an E_{0}-semigroup α on B(H̃) if for all A ∈ B(H̃) and all t ≥ 0,

U_{t}A = α_{t}(A)U_{t}.

Remark. We shall denote by U_{α} the family of units of α. It is not difficult to show that any U ∈ U_{α} is, up to perturbation by a semigroup of scalar operators, a one-parameter semigroup of isometries.

Definition 4.2. An E_{0}-semigroup α on B(H̃) is said to be completely spatial if for all t ≥ 0 the Hilbert space H̃ coincides with the closed linear span of all vectors in H̃ of the form

(4.4)  U_{s_{1}}^{j_{1}}U_{s_{2}}^{j_{2}} ⋯ U_{s_{n}}^{j_{n}}ξ,

with U^{j_{ℓ}} ∈ U_{α}, j = 1, 2, …, n, with ξ ∈ B(H̃), and with ∑_{r=1}^{n} s_{r} = t.

proof of theorem. Let α be a minimal dilation of ϕ. Since ϕ is a CP semigroup on a matrix algebra its minimal dilation is completely spatial, by [A6] or [P1]. For each k ∈ ℕ let α^{k} be the E_{0}-semigroup defined by α_{t}^{k} = α_{tλ_{k}} . α^{k} is clearly a minimal dilation of the CP semigroup ϕ^{k} related to ϕ by ϕ_{t}^{k} = ϕ_{tλ_{k}}. Hence each α^{k} is completely spatial.

Using the notation of section 2, for some m ∈ ℕ, let h = ξ_{1}⊗ξ_{2}⊗⋯⊗ξ_{m}⊗f_{m+1}⊗ f_{m+2}⊗⋯, where ξ_{k} ∈ H̃_{k} is arbitrary and f_{k}, k = 1, 2, … are the distinguished unit vectors used in the construction of the tensor product Hilbert space H. Since linear combinations of such vectors are dense in H we will be done if we can show that such vectors are in the closed linear span of vectors of the form (4.4). First note that the finite tensor product of completely spatial E_{0}-semigroups is completely spatial. One sees this by recalling that the finite tensor product of CAR flows is a CAR flow [P3] and then using the equivalence, up to cocycle conjugacy, between CAR flows and completely spatial semigroups, [A1]. ¿From this observation it follows that the product vector ξ_{1} ⊗ ξ_{2} ⊗ ⋯ ⊗ ξ_{m} lies in the closed linear span of vectors of the form

U_{s_{1}}^{j_{1}}U_{s_{2}}^{j_{2}} ⋯ U_{s_{n}}^{j_{n}}ξ, where each U^{j_{ℓ}} is a unit of α^{1} ⊗ ⋯ ⊗ α^{m} and ξ ∈ ⊗_{k=1}^{m}H_{k}.

Now let W be any fixed isometric unit of α. Note then that the semigroup W^{k} of isometries given by W_{t}^{k} = W_{tλ_{k}}, t ≥ 0 is a unit of α^{k}. We wish to make sense of the infinite tensor product W^{m+1} ⊗ W^{m+2} ⊗ ⋯ and to show it is a unit of the E_{0}-semigroup ⊗_{k=m+1}^{∞}α^{k}. To see this, let P be the orthogonal projection in B(H̃),

with range H, which compresses α̃ to ϕ̃. Then by [A4] C = {C_{t} = P W_{t}P : t ≥ 0} is a contraction semigroup on H. For any unit vector f in H,

∥W_{s}f − f∥^{2} = 2(∥f∥^{2} − ℜ ⟨W_{s}f, f⟩) = 2ℜ(∥f∥^{2} − ⟨C_{s}f, f⟩) ≤ 2|1 − ⟨C_{s}f, f⟩ |.

Using the fact that C_{s} = e^{sA} for a bounded operator A we have

|1 − ⟨C_{s}f, f⟩ | = | ∫_{0}^{s} \frac{d}{ds} ⟨C_{s}f, f⟩ ds| = | ∫_{0}^{s} ⟨C_{s}Af, f⟩ ds| ≤ ∥A∥ · s.

Taking square roots we arrive at the estimate

(E)  ∥W_{s}f − f∥ ≤ c · \sqrt{s},

where c is a constant. It will follow from this calculation that for fixed s ≥ 0 the sequence of operators V_{q}(s) = I ⊗ I ⊗ ⋯ ⊗ I ⊗ W_{s}^{m+1} ⊗ W_{s}^{m+2} ⊗ ⋯ ⊗ W_{s}^{q} ⊗ I ⊗ I ⊗ ⋯, q = m + 1, m + 2, …, converges strongly in B(H̃). To verify this it suffices to show, since the sequence of operators is bounded in norm, that the sequence {V_{q}(s)g : q ≥ m + 1} is Cauchy for each g in some dense subset H̃_{0} of H̃. We may take H̃_{0} to be the set of finite linear combinations of tensor products of unit vectors of the form g = g_{1} ⊗ g_{2} ⊗ ⋯ g_{r} ⊗ f_{r+1} ⊗ f_{r+2} ⊗ ⋯, r ∈ ℕ, g_{k} ∈ H̃_{k}, k = 1, 2, … r. It follows that it suffices to show that {V_{q}(s)g : q ≥ m + 1} is Cauchy for tensor product vectors of this form. Using (E) we have, for r < q_{1} < q_{2},

∥V_{q_{2}}(s)g − V_{q_{1}}(s)g∥ ≤ ∑_{k=q_{1}+1}^{q_{2}} ∥V_{k}(s)g − V_{k−1}(s)g∥

≤ ∑_{k=q_{1}+1}^{q_{2}} ∥W_{s}^{k}f_{k} − f_{k}∥

≤ ∑_{k=q_{1}+1}^{q_{2}} c · \sqrt{s} · \sqrt{λ_{k}}

This establishes our claim about the existence of the tensor product I ⊗ I ⊗ ⋯ I ⊗ ⊗_{k=m+1}^{∞}W^{k} in B(H̃_{1} ⊗ H̃_{2} ⊗ ⋯ H̃_{m})′ ∩ B(H̃) of the isometric units W^{k}. A similar

calculation to the one above shows that this semigroup is strongly continuous in the real variable s. Moreover, following the construction in section 2 it is not difficult to show that for fixed m ∈ ℕ, W̄ = W^{m+1} ⊗ W^{m+2} ⊗ ⋯ is a unit of ⊗_{k=m+1}^{∞}α^{k}.

Hence if U^{ℓ} is a unit of ⊗_{k=1}^{m}α^{k}, then U^{ℓ} ⊗ W̄ is a unit of ⊗_{k=1}^{∞}α^{k}.

Finally let h be the product vector given above, let t ≥ 0 be fixed, and for

ϵ > 0 let m sufficiently large that ∥W̄_{t}( ⊗_{k=m+1}^{∞}f_{k}) − ⊗_{k=m+1}^{∞}f_{k}∥ < ϵ/2. Since ⊗_{k=1}^{m}α^{k} is

completely spatial there are units U^{ℓ} in B(H̃_{1} ⊗ H̃_{2} ⊗ ⋯ H̃_{m}) of this E_{0}-semigroup such that some linear combination F of vectors of the form U_{s_{1}}^{j_{1}}U_{s_{2}}^{j_{2}} ⋯ U_{s_{n}}^{j_{n}}ξ satisfies ∥F − ξ_{1} ⊗ ξ_{2} ⊗ ⋯ ⊗ ξ_{m}∥ < ϵ/2. Noting that U^{ℓ} ⊗ W̄ is a unit of α̃ for all units U^{ℓ} of ⊗_{k=1}^{m}α^{k}, and noting that ∥(F ⊗ W̄_{t}(f_{m+1} ⊗ f_{m+2} ⊗ ⋯)) − h∥ < ϵ we have verified

that α̃ is completely spatial.

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