THE HEAT FLOW OF THE CCR ALGEBRA

William Arveson

Department of Mathematics University of California Berkeley CA 94720, USA

29 February, 2000

Abstract. Let P f(x) = −if′(x) and Qf(x) = xf(x) be the canonical operators acting on an appropriate common dense domain in L^{2}(ℝ). The derivations D_{P} (A) = i(P A−AP) and D_{Q}(A) = i(QA−AQ) act on the ∗-algebra A of all integral operators having smooth kernels of compact support, for example, and one may consider the noncommutative “Laplacian” L = D_{P}^{2} + D_{Q}^{2} as a linear mapping of A into itself.

L generates a semigroup of normal completely positive linear maps on B(L^{2}(ℝ)), and we establish some basic properties of this semigroup and its minimal dilation to an E_{0}-semigroup. In particular, we show that its minimal dilation is pure, has no normal invariant states, and in section 3 we discuss the significance of those facts for the interaction theory introduced in a previous paper.

There are similar results for the canonical commutation relations with n degrees of freedom, n = 2, 3, . . ..

1. Discussion, basic results.

Consider the canonical operators P, Q acting on an appropriate common dense domain in L^{2}(ℝ)

P = \frac{1}{i} · \frac{d}{dx},
Q = multiplication by x.

These operators can be used to define unbounded derivations (say on the dense ∗- algebra A of all integral operators having kernels which are smooth and of compact support) by

D_{P} (X) = i(P X − XP),  D_{Q}(X) = i(QX − XQ),  X ∈ A.

Thinking of these derivations as noncommutative counterparts of ∂/∂x and ∂/∂y we define a “Laplacian” L : A → A by

(1.1)  L = D_{P}^{2} + D_{Q}^{2}.

On appointment as a Miller Research Professor in the Miller Institute for Basic Research in Science. Support is also acknowledged from NSF grant DMS-9802474

Throughout this paper we will use the term CP semigroup to denote a semigroup ϕ = {ϕ_{t} : t ≥ 0} of normal completely positive linear maps on the algebra B(H) of all bounded operators on a separable Hilbert space H, which preserves the unit ϕ_{t}(1) = 1, and which is continuous in the natural sense (namely ⟨ϕ_{t}(A)ξ, η⟩ should be continuous in t for fixed ξ, η ∈ H and A ∈ B(H)). The purpose of this section is to exhibit concretely a CP semigroup whose generator can be identified with the operator mapping L of (1.1) (see Theorem 1.10).

Let U_{t} = e^{itQ}, V_{t} = e^{itP} be the two unitary groups associated with Q, P,

U_{t}f(x) = e^{itx}f(x),  V_{t}f(x) = f(x + t),  f ∈ L^{2}(ℝ).

These two groups satisfy the Canonical Commutation Relations V_{t}U_{s} = e^{ist}U_{s}V_{t} for s, t ∈ ℝ. It is more convenient to make use of the CCRs in Weyl’s form. For every z = (x, y) ∈ ℝ^{2} the Weyl operator

(1.2)  W_{z} = e^{\frac{ixy}{2}} U_{x}V_{y}

is unitary, it is strongly continuous in z, and it satisfies the Weyl relations

(1.3)  W_{z_{1}}W_{z_{2}} = e^{iω(z_{1},z_{2})}W_{z_{1}+z_{2}}

where ω is the symplectic form on ℝ^{2} given by

(1.4)  ω((x, y), (x′, y′)) = \frac{1}{2}(x′y − xy′).

A strongly continuous mapping z ↦ W_{z} ∈ B(H) into the unitary operators on some Hilbert space H which satisfies (1.3) is called a Weyl system. It is well known that the Weyl system (1.2) is irreducible, and hence the space of all finite linear combinations of the W_{z} is a unital strongly dense ∗-subalgebra of B(L^{2}(ℝ)). The Stone-von Neumann theorem implies that every Weyl system is unitarily equivalent to a direct sum of copies of the concrete Weyl system (1.2).

Proceeding heuristically for a moment, let D_{P} and D_{Q} be the derivations above. After formally differenting the relation V_{t}U_{s} = e^{ist}U_{s}V_{t} we find that

D_{P} (U_{x}) = ixU_{x},  D_{P} (V_{y}) = 0,
D_{Q}(U_{x}) = 0,  D_{Q}(V_{y}) = iyU_{y},

hence the action of L = D_{P}^{2} + D_{Q}^{2} on the Weyl system (1.2) is given by

L(W_{z}) = −(x^{2} + y^{2})W_{z} = −|z|^{2}W_{z}, z = (x, y) ∈ ℝ^{2}.

After formally exponentiating we find that for t ≥ 0 the operator mapping ϕ_{t} = exp(tL) for t ≥ 0 can be expected to satisfy

(1.5)  ϕ_{t}(W_{z}) = e^{−t|z|^{2}}W_{z},  z ∈ ℝ^{2}, t ≥ 0.

Remarks. A number of authors have considered completely positive semigroups defined on a Weyl system by formulas such as (1.5), using techniques similar to those of Proposition 1.7 below (see pp. 128-129 of [3], or [6] for two notable examples).

THE CCR HEAT FLOW 3

We include a full discussion of these basic issues since in section 3 we require details of the construction that are not easily found in the literature.

We also remark that virtually all of the results below have straightforward generalizations to the case in which P, Q are replaced with the canonical operators P_{1}, …, P_{n}, Q_{1}, …, Q_{n} associated with n degrees of freedom. Indeed, the generalization amounts to little more than a reinterpretation of notation. On the other hand, while it is possible to reformulate Proposition 1.7 in a form valid for infinitely many degrees of freedom (c.f. [3] loc cit), we do not know if that is the case for the more precise results of section 3.

In order to define the CCR heat flow rigorously we take (1.5) as our starting point and deduce the existence of the semigroup and its basic properties from the following general result. Consider the Banach space M(ℝ^{2}) of all complex-valued measures μ on ℝ^{2} having finite total variation ∥μ∥. M(ℝ^{2}) is a commutative Banach algebra with unit relative to the usual convolution of measures

μ ∗ ν(S) = ∫_{ℝ^{2}×ℝ^{2}} χ_{S}(z + w) dμ(z) dν(w).

It will be convenient to define the Fourier transform of a measure μ ∈ M(ℝ^{2}) in terms of the symplectic form ω of (1.4)

(1.6)  μ̂(ζ) = ∫_{ℝ^{2}} e^{iω(ζ,z)} dμ(z).

Remark. While this definition of the Fourier transform differs from the usual one, which involves the Euclidean inner product of ℝ^{2}

⟨(x, y), (x′, y′)⟩ = xx′ + yy′

rather than the symplectic form ω, it is equivalent to it in a natural way. Indeed, since ω is nondegenerate there is a unique invertible skew symmetric linear operator Ω on the two dimensional real vector space ℝ^{2} satisfying ω(z, z′) = ⟨Ωz, z′⟩ for all z, z′ ∈ ℝ^{2}. Hence one can pass back and forth from the usual Fourier transform of a measure to the one above by the invertible linear change-of-variables given by composing the transformed measure with either Ω or Ω^{−1} = −4Ω.

Proposition 1.7. Let {W_{z} : z ∈ ℝ^{2}} be an irreducible Weyl system acting on a Hilbert space H. For every complex measure μ ∈ M(ℝ^{2}) there is a unique normal completely bounded linear map ϕ_{μ} : B(H) → B(H) satisfying

ϕ_{μ}(W_{z}) = μ̂(z)W_{z}, z ∈ ℝ^{2}.

One has ϕ_{μ} ∘ ϕ_{ν} = ϕ_{μ∗ν}, and ∥ϕ_{μ}∥_{cb} ≤ ∥μ∥, where ∥ψ∥_{cb} denotes the completely bounded norm of an operator mapping ψ. When μ is a positive measure ϕ_{μ} is a completely positive map.

proof. Fix μ ∈ M(ℝ^{2}). Without loss of generality we may assume that ∥μ∥ = 1. The uniqueness of the mapping ϕ_{μ} is apparent from the irreducibility hypothesis on the Weyl system, since the set of all linear combinations of the W_{z}, z ∈ ℝ^{2}, is a unital ∗-algebra which is weak^{∗}-dense in B(H).

To prove existence, note first that there is a strongly continuous unitary representation U of the additive group ℝ^{2} on a separable Hilbert space K and a pair of unit vectors f, g ∈ K such that

(1.8)  μ̂(ζ) = ⟨U_{ζ}f, g⟩ ,  ζ ∈ ℝ^{2}.

To see this, notice that there is a probability measure |μ| and a measurable function v : ℝ^{2} → ℂ satisfying |v(z)| = 1 almost everywhere (d|μ|) such that

μ(S) = ∫_{S} v(z) d|μ|(z)

for every Borel set S ⊆ ℝ^{2} (i.e., the pair (v, |μ|) is simply the “polar decomposition” of μ). Let K = L^{2}(ℝ^{2}, |μ|) and define unit vectors f, g in K by f(z) = v(z) and g(z) = 1. We can define a unitary representation U on L^{2}(ℝ^{2}, |μ|) by setting

U_{ζ}h(z) = e^{iω(ζ,z)}h(z), h ∈ L^{2}(ℝ^{2}, |μ|)

and the asserted formula (1.8) follows immediately.

Choosing U : ℝ^{2} → B(K) and unit vectors f, g ∈ K so that (1.8) is satisfied, consider the Hilbert space K ⊗H and the slice map σ_{μ} : B(K ⊗H) → B(H) defined by

⟨σ_{μ}(T)ξ, η⟩ = ⟨T (f ⊗ ξ), g ⊗ η⟩ , ξ, η ∈ H, T ∈ B(K ⊗ H).

Since f and g are unit vectors, the completely bounded norm of σ_{μ} is at most 1. Taking T = U_{ζ} ⊗ W_{ζ} we find that

(1.9)  σ(U_{ζ} ⊗ W_{ζ}) = μ̂(ζ)W_{ζ},  ζ ∈ ℝ^{2}.

Now consider W′_{ζ} = U_{ζ} ⊗ W_{ζ} ∈ B(K ⊗ H). W′ is obviously a Weyl system, and by the Stone-von Neumann theorem it must be unitarily equivalent to a direct sum of copies of the irreducible Weyl system W acting on H. Thus there is a unique (normal) ∗-homomomorphism of von Neumann algebras θ : B(H) → B(K ⊗ H) such that θ(W_{ζ}) = U_{ζ} ⊗ W_{ζ} for all ζ ∈ ℝ^{2}. Defining ϕ_{μ} to be the composition σ_{μ} ∘ θ, we find that ϕ_{μ} is a normal operator mapping of completely bounded norm at most 1, for which ϕ_{μ}(W_{ζ}) = μ̂(ζ)W_{ζ}.

If μ happens to be a positive measure then the function v of the polar decomposition of μ can be taken as the constant function 1, hence formula (1.8) reduces to μ̂(ζ) = ⟨U_{ζ}f, f⟩, i.e., g = f. From this it is apparent that the corresponding slice map σ_{μ} above is completely positive, and hence ϕ_{μ} = σ ∘ θ is completely positive.

The semigroup property ϕ_{μ} ∘ ϕ_{ν} = ϕ_{μ∗ν} follows immediately from the defining relations ϕ_{μ}(W_{z}) = μ̂(z)W_{z}, ϕ_{ν}(W_{z}) = ν̂(z)W_{z}, and the fact that (μ ∗ ν)ˆ (z) = μ̂(z)ν̂(z) for z ∈ ℝ^{2}.

Theorem 1.10. Let W = {W_{z} : z ∈ ℝ^{2}} be an irreducible Weyl system. Then there is a unique CP semigroup ϕ = {ϕ_{t} : t ≥ 0} satisfying

(1.11)  ϕ_{t}(W_{z}) = e^{−t|z|^{2}}W_{z},  z ∈ ℝ^{2}.

The only bounded normal linear functional ρ for which ρ ∘ ϕ_{t} = ρ for all t ≥ 0 is ρ = 0. In particular, there is no normal state of B(H) which is invariant under ϕ.

proof. For each t ≥ 0, u_{t}(z) = e^{−t|z|^{2}} is a continuous function of positive type, which takes the value 1 at z = 0. Thus it is the Fourier transform of a unique probability measure μ_{t} ∈ M(ℝ^{2}). We will require an explicit formula for the Gaussian measure μ_{t} later on; but for purposes of this section we require nothing more than its existence and uniqueness.

Since u_{s}(z)u_{t}(z) = u_{s+t}(z) for all z ∈ ℝ^{2} it follows that μ_{s} ∗ μ_{t} = μ_{s+t}. Hence Proposition 1.7 implies that there is a semigroup ϕ = {ϕ_{t} : t ≥ 0} of normal completely positive maps on B(H) which satisfies (1.11). It is a simple matter to check that the required continuity of ϕ_{t} in t follows from the continuity of the right side of (1.11) in t for fixed z.

Suppose now that ρ is a normal linear functional which is invariant under ϕ. Then for every z ∈ ℝ^{2} and every t ≥ 0, the definition of ϕ_{t} implies that

ρ(W_{z}) = ρ(ϕ_{t}(W_{z})) = e^{−t|z|^{2}}ρ(W_{z})

and for fixed z ≠ 0, the right side tends to 0 as t → ∞. Hence ρ(W_{z}) = 0 for every z ≠ 0; by strong continuity on the unit ball it follows that ρ(1) = ω(W_{0}) = 0. hence ρ vanishes on the irreducible ∗-algebra spanned by W_{z}, z ∈ ℝ^{2} and by normality it follows that ρ = 0.

Remarks. We point out that while ϕ has no normal invariant states, it does have a normal invariant weight...namely the trace, in that

trace (ϕ_{t}(A)) = trace (A)

for every positive operator A ∈ B(L^{2}(ℝ)) and every t ≥ 0. We omit the proof since we do not require this result.

We also remark that one can deduce the existence of other CP semigroups along similar lines. For example, the proof of Theorem 1.10 implies that there is a “Cauchy” semigroup ψ = {ψ_{t} : t ≥ 0} which is defined uniquely by the requirement

ψ_{t}(W_{z}) = e^{−t(|x|+|y|)}W_{z}, z = (x, y) ∈ ℝ^{2}.

2. Harmonic analysis of the commutation relations.

A classical theorem of Beurling asserts that singletons obey spectral synthesis. More precisely, if G is a locally compact abelian group and f is an integrable function on G whose Fourier transform vanishes at a point p in the dual of G, then there is a sequence of functions f_{n} ∈ L^{1}(G) such that ∥f − f_{n}∥ → 0 as n → ∞ and such that the Fourier transform of each f_{n} vanishes identically on some open neighborhood U_{n} of p. The purpose of this section is to present a noncommutative version of that result, which will be required in section 3.

Let {W_{z} : z ∈ ℝ^{2}} be an irreducible Weyl system acting on a Hilbert space H (for example, one may take the Weyl system (1.2) acting on L^{2}(ℝ^{2})). For every trace-class operator A ∈ L^{1}(H) we consider the following analogue of the Fourier transform Â : ℝ̂^{2} → ℂ

Â(z) = trace(AW_{z}), z ∈ ℝ^{2}.

This transform A ∈ L^{1}(H) ↦ Â shares many features in common with the commutative Fourier transform. For example, using the concrete realization (1.2), it is quite easy to establish a version of the Riemann-Lebesgue lemma

lim_{|z|→∞} Â(z) = 0,

for every A ∈ L^{1}(H). What we actually require is the following analogue of Beurling’s theorem, which lies somewhat deeper.

Theorem 2.1. Let A ∈ L^{1}(H) and let ζ ∈ ℝ^{2} be such that trace(AW_{ζ}) = 0. There is a sequence A_{n} ∈ L^{1}(H) and a sequence of open neighborhoods U_{n} of ζ such that

trace(A_{n}W_{z}) = 0, z ∈ U_{n},

and such that trace|A − A_{n}| → 0 as n → ∞.

proof. By replacing A with AW_{ζ} and making obvious use of the canonical commutation relations (1.3), we may immediately reduce to the case ζ = 0. We find it more convenient to establish the dual assertion of Theorem 2.1. For that, consider the following linear subspaces of B(H)

S_{ϵ} = \overline{span}\{W_{z} : |z| ≤ ϵ\}, ϵ > 0,

the closure being taken relative to the weak^{∗} topology on B(H). Obviously the spaces S_{ϵ} decrease as ϵ decreases, and the identity operator belongs to S_{ϵ} for every ϵ > 0. The pre-annihilator of S_{ϵ} is identified with the space of all trace-class operators A satisfying

(2.2)  Â(z) = trace(AW_{z}) = 0,  |z| ≤ ϵ.

Lemma 2.3. Let {W_{z} : z ∈ ℝ^{2}} be an arbitrary Weyl system acting on a separable Hilbert space H. Then ∩{S_{ϵ} : ϵ > 0} = ℂ · 1.

proof of Lemma 2.3. Let S_{0} denote the intersection ∩{S_{ϵ} : ϵ > 0}. We have already remarked that the inclusion ⊇ is obvious. For the opposite one, consider the von Neumann algebra M generated by {W_{z} : z ∈ ℝ^{2}}. M is a factor (of type I_{∞}) because of the Stone-von Neumann theorem. We will show that S_{0} is contained in the center of M.

For that, choose T ∈ S_{0} and consider the operator-valued function z ↦ W_{z}T W_{z}^{∗}. We have to show that this function is constant; equivalently, we will show that for fixed ξ and η in H, the function

(2.4)  z ∈ ℝ^{2} ↦ ⟨W_{z}T W_{z}^{∗}ξ, η⟩

is constant. Since the function of (2.4) is bounded and continuous, it suffices to show that its spectrum (in the sense of spectral synthesis for functions in L^{∞}(ℝ^{2})) is the singleton {0}: this is the dual formulation of Beurling’s theorem cited above. Thus we have to show that for every function f ∈ L^{1}(ℝ^{2}) whose Fourier transform

f̂(ζ) = ∫_{ℝ^{2}} e^{iω(z,ζ)}f(z) dz

vanishes throughout a neighborhood of the origin ζ = 0, we have

(2.5)  ∫_{ℝ^{2}} f(z) ⟨W_{z}T W_{z}ξ, η⟩ dz = 0.

Fix such an f ∈ L^{1}(ℝ^{2}) and choose ϵ > 0 small enough so that f̂(ζ) = 0 for all ζ satisfying |ζ| ≤ ϵ. Since the linear functional

X ∈ B(H) ↦ ∫_{ℝ^{2}} f(z) ⟨W_{z}XW_{z}^{∗}ξ, η⟩ dz

is weak^{∗}-continuous and T belongs to the weak^{∗}-closed linear span of operators of the form W_{ζ} with |ζ| ≤ ϵ, to prove (2.5) it suffices to show that for every ζ with |ζ| ≤ ϵ we have

(2.6)  ∫_{ℝ^{2}} f(z) ⟨W_{z}W_{ζ}W_{z}^{∗}ξ, η⟩ dz = 0.

Using the canonical commutation relations we can write

W_{z}W_{ζ}W_{z}^{∗} = e^{iω(z,ζ)}W_{z+ζ}W_{−z} = e^{ω(ζ,−z)}W_{ζ} = e^{iω(z,ζ)}W_{ζ}.

Hence the left side of (2.6) becomes

∫_{ℝ^{2}} f(z)e^{iω(z,ζ)} ⟨W_{ζ}ξ, η⟩ dz = f̂(ζ) ⟨W_{ζ}ξ, η⟩ ,

and the latter term vanishes because f̂(ζ) = 0 for |ζ| ≤ ϵ.

To complete the proof of Theorem 2.1, choose an operator A ∈ L^{1}(H) satisfying

Â(0) = trace(A) = 0,

and consider the linear functional ρ defined on B(H) by ρ(T) = trace(AT). ρ obviously vanishes on ℂ·1. The linear spaces S_{ϵ} are weak^{∗}-closed and they decrease to S_{0} = ℂ · 1 as ϵ decreases to 0, by Lemma 2.3. Since ρ is weak^{∗}-continuous we must have

lim_{ϵ→0} ∥ρ ↾_{S_{ϵ}} ∥ = ∥ρ ↾_{ℂ·1} ∥ = 0.

Thus we can choose a sequence ϵ_{n} ↓ 0 so that ∥ρ ↾_{S_{ϵn}} ∥ ≤ 1/n for every n = 1, 2, . . .. We have already pointed out that the pre-annihilator of S_{ϵ_{n}} is identified with all trace class operators B satisfying

(2.7)  B̂(z) = trace(BW_{z}) = 0,  |z| ≤ ϵ_{n}.

Since ∥ρ ↾_{S_{ϵn}} ∥ is the trace-norm distance from A to the pre-annihilator of S_{ϵ_{n}}, we conclude that there is a sequence of operators B_{n} ∈ L^{1}(H) which satisfy trace(B_{n}W_{z}) = 0 for |z| ≤ ϵ_{n}, such that trace|A − B_{n}| ≤ 2/n, as asserted.

3. Purity and dilation theory.

An E_{0}-semigroup is a CP semigroup α = {α_{t} : t ≥ 0}, acting on B(H), such the the individual maps are endomorphisms, α_{t}(AB) = α_{t}(A)α_{t}(B), A, B ∈ B(H). An E_{0}-semigroup α is called pure if its “tail” von Neumann algebra is trivial,

(3.1)  ⋂_{t≥0} α_{t}(B(H)) = ℂ · 1.

It is known that an E_{0}-semigroup is pure iff for any pair of normal states ρ_{1}, ρ_{2} of B(H) we have

(3.2)  lim_{t→∞} ∥ρ_{1} ∘ α_{t} − ρ_{2} ∘ α_{t}∥ = 0

see [1].

If a pure E_{0}-semigroup α has a normal invariant state ω, then the characterization (3.2) implies that ω must be an absorbing state in the sense that for every normal state ρ of B(H) one has

(3.3)  lim_{t→∞} ∥ρ ∘ α_{t} − ω∥ = 0.

Conversely, if for an arbitrary E_{0}-semigroup α there is a state ω of B(H) which is absorbing in the sense that (3.3) is satisfied for every normal state ρ of B(H), then ω must be a normal invariant state, and thus by (3.2) α must be a pure E_{0}-semigroup.

In the theory of interactions worked out in [2], pure E_{0}-semigroups occupy a central position, especially those for which there is a normal invariant (and therefore absorbing) state. A natural question that emerges from the theory of interactions is whether or not every pure E_{0}-semigroup must have a normal invariant state. Now since the state space of B(H) is weak^{∗}-compact, a routine application of the Markov-Kakutani fixed point theorem shows shows that every E_{0}-semigroup must have invariant states; but invariant states obtained by such methods need not be normal. In this section we exhibit a concrete E_{0}-semigroup which is pure but which has no normal invariant states. This is a result which was asserted (without proof) in [2]. This E_{0}-semigroup is obtained from the CP semigroup of Theorem 1.10 by a dilation procedure.

In order that the minimal dilation of a CP semigroup to an E_{0}-semigroup should satisfy (3.1), it is necessary and sufficient that the CP semigroup should satisfy property (3.2) (see Proposition 3.5). Thus we generalize the definition of pure E_{0}-semigroup as follows.

Definition 3.4. A CP semigroup ϕ acting on B(H) is called pure if for every pair of normal states ρ_{1}, ρ_{2} of B(H) we have

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

Proposition 3.5. Let ϕ = {ϕ_{t} : t ≥ 0} be a pure CP semigroup which has no normal invariant state, and let α be its minimal dilation to an E_{0}-semigroup. Then α satisfies (3.1) and has no normal invariant state.

proof. The proof is straightforward, but we require results from [1]. Proposition 2.4 of [1] implies that α satisfies (3.1).

To see that α has no normal invariant state, we can assume that α acts on B(H) for some Hilbert space H and that there is a closed subspace K ⊆ H such that ϕ is the compression of α onto B(K) = P B(H)P, P denoting the projection of H onto K. We have α_{t}(P) ↑ 1 because α is minimal over P. So if ω is any normal state of B(H) which is invariant under α then we have

ω(P) = lim_{t→∞} ω(α_{t}(P)) = ω(1) = 1.

Thus the restriction of ω to B(K) = P B(H)P defines a normal ϕ-invariant state on B(K), contradicting the hypothesis on ϕ.

In the remainder of this section we show that the CP semigroup defined in Theorem 1.10 is pure. Once that is established, Proposition 3.5 implies that its minimal dilation is an E_{0}-semigroup with properties asserted in the discussion above.

Theorem 3.6. The CP semigroup ϕ defined in (1.11) is pure.

Before giving the proof, we require

Lemma 3.7. For each t > 0 let μ_{t} be the Gaussian measure on ℝ^{2} whose Fourier transform (1.6) is given by

μ̂_{t}(z) = e^{−t|z|^{2}}, z ∈ ℝ^{2},

and choose δ > 0. There is a family ν_{t}, t > 0, of probability measures on ℝ^{2} such that

(i) ν̂_{t}(z) = 0 for all |z| ≥ δ and every t, and (ii) lim_{t→∞} ∥μ_{t}−ν_{t}∥ = 0, ∥·∥ denoting the norm of the measure algebra M(ℝ^{2}).

proof. For t > 0, μ_{t} is given by dμ_{t} = u_{t}(x, y) dx dy, where u_{t} is the density

u_{t}(x, y) = \frac{1}{πt}e^{− \frac{x^{2}+y^{2}}{4t}} .

Let f_{t} = \sqrt{u_{t}}. f_{t} belongs to L^{2}(ℝ^{2}) and ∥f_{t}∥_{2} = 1. Choose a function g ∈ L^{1}(ℝ^{2}) whose Fourier transform

ĝ(ζ) = ∫_{ℝ^{2}} e^{iω(ζ,z)}g(z) dz

satisfies 0 ≤ ĝ(ζ) ≤ 1 for all ζ, and

ĝ(ζ) = \{ \begin{matrix}1, & for 0 ≤ |ζ| ≤ δ/4 \\ 0, & for |ζ| ≥ δ/2. \end{matrix}

Consider the convolution g ∗ f_{t} ∈ L^{2}(ℝ^{2}) and the positive measure

dν_{t} = |g ∗ f_{t}|^{2} dx dy.

ν_{t} is obviously a positive finite measure. We claim first that the Fourier transform of ν_{t} lives in the disk |ζ| ≤ δ. Indeed, letting U_{ζ} (resp. T_{ζ}) be the unitary operator on L^{2}(ℝ^{2}) (resp. L^{2}(ℝ̂^{2})) given by

U_{ζ}F (z) = e^{iω(ζ,z)}F (z), T_{ζ}G(w) = G(w + ζ),

we have by the Plancherel theorem

ν̂_{t}(ζ) = ⟨U_{ζ}(g ∗ f_{t}), g ∗ f_{t}⟩_{L^{2}(ℝ^{2})} = ⟨T_{ζ}(ĝf̂_{t}), ĝf̂_{t}⟩_{L^{2}(ℝ̂^{2})}
= ∫ℝ̂^{2}(ĝf̂_{t})(w + ζ)\overline{(ĝf̂_{t})}(w) dw.

When |ζ| ≥ δ the integrand on the right vanishes identically in w because ĝf̂_{t} is supported in the disk of radius δ/2. Hence ν̂_{t}(ζ) = 0 for |ζ| ≥ δ.

To establish property (ii), it is enough to show that

(3.8)  lim_{t→∞} ∥f_{t} − g ∗ f_{t}∥_{2} = 0,

since by the Schwarz inequality

∥μ_{t} − ν_{t}∥ = ∫_{ℝ^{2}} |f_{t}^{2} − |g ∗ f_{t}|^{2}| dz ≤ ∫_{ℝ^{2}} |f_{t} − g ∗ f_{t}| · |f_{t} + |g ∗ f_{t}|| dz

≤ ∥f_{t} − g ∗ f_{t}∥_{2} · ∥f_{t} + |g ∗ f_{t}|∥_{2} ≤ ∥f_{t} − g ∗ f_{t}∥_{2}(∥f_{t}∥_{2} + ∥g ∗ f_{t}∥_{2}).

To establish (3.8), we use the Plancherel theorem again to write

∫_{ℝ^{2}} |f_{t}(z) − g ∗ f_{t}(z)|^{2} dz = ∫ℝ̂^{2} |f̂(ζ) − ĝ(ζ)f̂_{t}(ζ)|^{2} dζ = ∫_{ℝ^{2}} |1 − ĝ(ζ)|^{2} · |f̂_{t}|^{2} dζ.

The function |1 − ĝ(ζ)| is bounded above by 1 and it vanishes throughout the disk 0 ≤ |ζ| ≤ δ/4. Hence the term on the right is dominated by

(3.9)  ∫_{{|ζ|≥δ/4}} |f̂_{t}(ζ)|^{2} dζ.

In order to estimate the integral (3.9) we require the explicit formula

f_{t}(x, y) = \sqrt{u_{t}(x, y)} = \frac{1}{\sqrt{πt}}e^{− \frac{x^{2}+y^{2}}{8t}} .

The Fourier transform of f_{t} has the form

f̂_{t}(ζ) = K\sqrt{t}e^{−2t|ζ|^{2}}

where K is a positive constant, hence (3.9) evaluates to

K^{2}t ∫_{{|ζ|≥δ/4}} e^{−4t|ζ|^{2}} dζ = K^{2} ∫_{S_{t}} e^{−4(u^{2}+v^{2})} du dv,

where S_{t} = \{(u, v) : \sqrt{u^{2} + v^{2}} ≥ (δ/4)\sqrt{t}\}. As t → ∞ the sets S_{t} decrease to ∅, hence the right side of the previous expression tends to 0, and (3.8) is proved.

The positive measures ν_{t} are not necessarily probability measures, but in view of the established property (ii), ν_{t}(ℝ^{2}) must be arbitrarily close to μ_{t}(ℝ^{2}) = 1 when t is large. Hence we can rescale ν_{t} in an obvious way to achive ν_{t}(ℝ^{2}) = 1 for all t > 0 as well as the properties (i) and (ii) of Lemma 3.7. proof of Theorem 3.6. Let W_{z}, z ∈ ℝ^{2} be an irreducible Weyl system acting on a Hilbert space H, and let ϕ = {ϕ_{t} : t ≥ 0} be the CP semigroup defined by the condition

ϕ_{t}(W_{z}) = e^{−t|z|^{2}}W_{z}, z ∈ ℝ^{2}.

Choose a pair of normal states ρ_{1}, ρ_{2} on B(H), and consider their difference ω = ρ_{1} − ρ_{2}. We have to show that

(3.10)  lim_{t→∞} ∥ω ∘ ϕ_{t}∥ = 0.

For that, let A be the self-adjoint trace-class operator defined by trace(AT) = ω(T), T ∈ B(H) and choose ϵ > 0. A has trace zero, so by Theorem 2.1, we can find a self-adjoint trace-class operator A_{0} such that trace(A_{0}W_{z}) = 0 for every z in some neighborhood U of z = 0, and trace|A − A_{0}| ≤ ϵ. It follows that the normal linear functional ω_{0}(T) = trace(A_{0}T) satisfies ∥ω − ω_{0}∥ ≤ ϵ and ω_{0}(W_{z}) = 0 for z ∈ U.

By Lemma 3.5 we can find probability measures ν_{t}, t > 0, such that ν̂_{t}(z) vanishes for z ∉ U and ∥μ_{t} − ν_{t}∥ tends to 0 as t → ∞. For each t > 0 let ψ_{t} be the completely positive map defined by Proposition 1.7,

ψ_{t}(W_{z}) = ν̂_{t}(z)W_{z}, z ∈ ℝ^{2}.

In order to prove (3.10) we decompose the linear functional ω ∘ ϕ_{t} into a sum of three terms as follows

(3.11)  ω ∘ ϕ_{t} = (ω − ω_{0}) ∘ ϕ_{t} + ω_{0} ∘ (ϕ_{t} − ψ_{t}) + ω_{0} ∘ ψ_{t}.

The third term on the right of (3.11) is zero because for every z ∈ ℝ^{2} we have

ω_{0}(ψ_{t}(W_{z})) = ω_{0}(ν̂_{t}(z)W_{z}) = ν̂_{t}(z)ω_{0}(W_{z}) = 0,

since ν̂_{t}(z) vanishes when z ∉ U and ω_{0}(W_{z}) vanishes when z ∈ U (recall that the linear span of the W_{z} for z ∈ ℝ^{2} is a strongly dense ∗-subalgebra of B(H)). The first term on the right of (3.11) is estimated for arbitrary t by

∥(ω − ω_{0}) ∘ ϕ_{t}∥ ≤ ∥ω − ω_{0}∥ ≤ ϵ.

In order to estimate the second term, note that

(3.12)  ∥ϕ_{t} − ψ_{t}∥ ≤ ∥μ_{t} − ν_{t}∥

for every t > 0. Indeed, considering the measure σ_{t} = μ_{t} − ν_{t} ∈ M(ℝ^{2}), we can write

ϕ_{t}(W_{z}) − ψ_{t}(W_{z}) = μ̂_{t}(z)W_{z} − ν̂_{t}(z)W_{z} = σ̂_{t}(z)W_{z}.

It follows from Proposition 1.7 that the completely bounded norm of the operator mapping ϕ_{t} − ψ_{t} is at most ∥σ_{t}∥ = ∥μ_{t} − ν_{t}∥, hence (3.12).

From (3.11) and these estimates we may conclude that

lim sup_{t→∞} ∥ω ∘ ϕ_{t}∥ ≤ ϵ + lim_{t→∞} ∥μ_{t} − ν_{t}∥ + 0 = ϵ.

Since ϵ is arbitrary the limit (3.10) is proved.

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