ON THE INDEX AND DILATIONS OF

COMPLETELY POSITIVE SEMIGROUPS

William Arveson

Department of Mathematics University of California Berkeley CA 94720, USA

28 August 1996

Abstract. It is known that every semigroup of normal completely positive maps P = {P_{t} : t ≥ 0} of B(H), satisfying P_{t}(1) = 1 for every t ≥ 0, has a minimal dilation to an E_{0}-semigroup acting on B(K) for some Hilbert space K ⊇ H. The minimal dilation of P is unique up to conjugacy. In a previous paper a numerical index was introduced for semigroups of completely positive maps and it was shown that the index of P agrees with the index of its minimal dilation to an E_{0}-semigroup. However, no examples were discussed, and no computations were made.

In this paper we calculate the index of a unital completely positive semigroup whose generator is a bounded operator

L : B(H) → B(H)

in terms of natrual structures associated with the generator. This includes all unital CP semigroups acting on matrix algebras. We also show that the minimal dilation of the semigroup P = {exp tL : t ≥ 0} to an E_{0}-semigroup is is cocycle conjugate to a CAR/CCR flow.

Introduction.

In [4], a numerical index is introduced for semigroups P = {P_{t} : t ≥ 0} of normal completely positive maps of B(H). In the case where P_{t}(1) = 1 for every t, a recent theorem of B. V. R. Bhat asserts that P can be “dilated” to an E_{0}- semigroup [5,6]; and it was shown in [4] that the index of P agrees with the index of its minimal E_{0}-semigroup dilation. However, no examples were discussed there. In particular, the results of [4] give no information about which E_{0}-semigroups can occur as the minimal dilations of unital completely positive semigroups acting on matrix algebras.

In this paper we consider the more general case of completely positive semigroups having bounded generator. We calculate the index of such semigroups in terms of basic structures associated with their generators (Theorem 2.3, Corollary 2.17) and

1991 Mathematics Subject Classification. Primary 46L40; Secondary 81E05. Key words and phrases. index, quantum dynamical semigroups, E_{0}-semigroups, von Neumann

algebras, completely positive maps.

This research was supported by NSF grant DMS95-00291

Typeset by AMS-TEX

in the case where the semigroup preserves the unit we show that their minimal dilations must be cocycle conjugate to a CAR/CCR flow (Corollary 4.21). The extent to which the index calculations of section 2 can be extended to semigroups with unbounded generators remains unclear at present; but certainly the description of their minimal dilations (e.g., Corollary 4.21) becomes false without strong hypotheses on the generator.

It is appropriate to point out that, using a completely different method, Powers [11] has independently shown that every unital completely positive semigroup acting on a matrix algebra dilates to a completely spatial E_{0}-semigroup and he calculates the index of the minimal dilation in that case.

1. Bounded generators, symbols, and metric operator spaces. We are concerned with the structure of various linear mappings on the von

Neumann algebra M = B(H), H being a separable Hilbert space. L(M) will denote the space of all bounded linear maps L : M → M. The purpose of this section is to discuss the relationship of metric operator spaces [4] to the generators of completely positive semigroups and their symbols. Our methods in §§2–3, even the statement of the key Theorem 2.3, will involve metric operator spaces in an essential way.

We briefly recall the definition of the symbol of a linear map L ∈ L(M). Consider the bilinear mapping defined on M × M by

(1.1)  L(xy) − xL(y) − L(x)y + xL(1)y.

It is useful to regard this as a homomorphism of the bimodule Ω^{2} of all noncommutative 2-forms into M, and that homomorphism of M-modules is the symbol of L. More explicitly, Ω^{1} is defined as the submodule of the symmetric bimodule M ⊗ M (with operations a(x ⊗ y)b = ax ⊗ yb, (x ⊗ y)^{∗} = y^{∗} ⊗ x^{∗}) generated by the range of the derivation d : M → M ⊗ M,

dx = x ⊗ 1 − 1 ⊗ x.

We have (dx)^{∗} = −d(x^{∗}) and every element of Ω^{1} is a finite sum of the form adx_{1} + ⋯ + adx_{n}, a_{k}, x_{k} ∈ M. Ω^{2} is defined by

Ω^{2} = Ω^{1} ⊗_{M} Ω^{1}.

Every element of Ω^{2} is a sum of the form a_{1}dx_{1} dy_{1} + ⋯ + a_{n}dx_{n} dy_{n}, and we have a natural multiplication

ω_{1}, ω_{2} ∈ Ω^{1} → ω_{1}ω_{2} ∈ Ω^{2}

which satisfies the associative law ω_{1}(aω_{2}) = (ω_{1}a)ω_{2} with respect to operators a ∈ M.

Given L ∈ L(M) there is a unique σ_{L} ∈ hom(Ω^{2}, M) satisfying

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

(see [3]). σ_{L} is called the symbol of L, and it has the following basic properties: (1.2) σ_{L} = 0 iff L has the form L(x) = ax + xb for fixed elements a, b ∈ M. (1.3) If σ_{L} = 0 and L satisfies L(x^{∗}) = L(x)^{∗} for every x, then there is an element

a ∈ M such that L(x) = ax + xa^{∗}. (1.4) For every x, y ∈ M we have

∥σ_{L}(dx dy)∥ ≤ 4∥L∥ ∥x∥ ∥y∥.

Remark 1.5. Property (1.2) follows from the fact that if σ_{L} = 0 then the linear map L_{0}(x) = L(x)−xL(1) is a derivation of M, and hence has the form L_{0}(x) = ax−xa. Property (1.3) follows from (1.2) after an elementary argument which uses the fact that if c is any operator satisfying cx + xc^{∗} = 0 for every operator x then c must have the form c = \sqrt{−1}λ1 where λ ∈ ℝ (see Lemma 1.19 below).

The following proposition summarizes some known results that illustrate how properties of the symbol characterize the generators of semigroups of completely positive maps.

Proposition 1.6. Suppose that A is a unital C^{∗}-algebra and L ∈ L(A) is a bounded operator satisfying L(x)^{∗} = L(x^{∗}), x ∈ A. Let {P_{t} : t ≥ 0} be the semigroup of linear operators on A obtained by exponentiation: P_{t} = exp(tL). Then the following are equivalent. 1.6.1 Each map P_{t} is completely positive. 1.6.2 If a_{1}, x_{1}, …, a_{n}, x_{n} ∈ A satisfy x_{1}a_{1} + … x_{n}a_{n} = 0 then we have

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

1.6.3 σ_{L}(ω^{∗}ω) ≤ 0 for all ω_{1}, ω_{2} ∈ Ω^{1}.

proof. The equivalence of (1.6.1) and (1.6.2) is essentially a result of Evans and Lewis [8]. The equivalence of (1.6.2) and (1.6.3) is discussed in [3]. □

It is a straightforward consequence of Stinespring’s theorem that every normal completely positive linear map P ∈ L(M) can be expressed in the form

(1.7)  P (x) = ∑_{k} v_{k}xv_{k}^{∗}

where {v_{1}, v_{2}, … } is a (finite or infinite) sequence of operators in M. Since certain facts relating to this representation are fundamental to our approach, we offer the following comments. Let (π, V) be a pair consisting of a representation π of M = B(H) on some other Hilbert space space K and an operator V ∈ B(H, K) satisfying

(1.8)  P (x) = V^{∗}π(x)V.

By cutting down to a subspace of K if necessary we can assume that K is spanned by {π(x)ξ : x ∈ M, ξ ∈ H}, and in this case the normality of P implies that π is a normal representation, necessarily nondegenerate. Thus by replacing (π, V) by an equivalent pair we may assume that K = H^{n} is a countable direct sum of copies of H and π has the form π(x) = x ⊕ x ⊕ . . .. It follows that there is a sequence v_{1}, v_{2}, ⋯ ∈ B(H) such that

V ξ = (v_{1}ξ, v_{2}ξ, …)

and the representation (1.7) follows by taking u_{k} = v_{k}^{∗}.

There is a natural operator space E_{P} associated with P, which can be defined in concrete terms as follows. Notice that if λ = (λ_{1}, λ_{2}, …) is any sequence in ℓ^{2} then the operator sum ∑_{k} λ_{k}u_{k} is convergent in the strong operator topology (this sum represents the composition of V^{∗} with the operator ξ ∈ H ↦ (λ_{1}ξ, λ_{2}ξ, …) ∈ H^{n}), and because of the minimality of (π, V) we have

(1.9)  λ_{1}u_{1} + λ_{2}u_{2} ⋯ = 0 ⟹ λ_{1} = λ_{2} = ⋯ = 0,

for every λ ∈ ℓ^{2}. We define

(1.10)  E_{P} = {λ_{1}u_{1} + λ_{2}u_{2} + ⋯: λ ∈ ℓ^{2}}.

E_{P} is not necessarily closed in the operator norm when it is infinite dimensional, but in all cases it is a Hilbert space with respect to the inner product defined on it by declaring {u_{1}, u_{2}, … } to be an orthonormal basis. E_{P} has the following properties. (1.11.1) An operator a belongs to E_{P} if and only if there is a constant c ≥ 0 such

that the mapping

x ∈ M ↦ cP (x) − axa^{∗}

is completely positive. In this case, ⟨a, a⟩_{E} is the smallest such constant c. (1.11.2) If w_{1}, w_{2}, … is any orthonormal basis for E_{P} then the sum ∑_{k} w_{k}w_{k}^{∗} con-

verges strongly, and in fact

P (x) = ∑_{k} w_{k}xw_{k}^{∗}, x ∈ M.

(1.11.3) If z_{1}, z_{2}, … is any finite or infinite sequence of operators in M such that the series ∑_{k} z_{k}z_{k}^{∗} converges strongly, and which represents P in the sense that

P (x) = ∑_{k} z_{k}xz_{k}^{∗}, x ∈ M,

then {z_{1}, z_{2}, … } spans the Hilbert space E_{P}. If, in addition, z_{1}, z_{2}, … satisfies the linear independence condition (1.9), then it is an orthonormal basis for E_{P}.

These properties are discussed more fully in [4].

Definition 1.12. A metric operator space is a pair (E, ⟨·, ·⟩) consisting of a linear supspace E ⊆ M = B(H) and an inner product ⟨·, ·⟩ : E × E → ℂ with respect to which E is a separable Hilbert space with the following property: for any orthonormal basis v_{1}, v_{2}, … for E we have

∥v_{1}^{∗}ξ∥^{2} + ∥v_{2}^{∗}ξ∥^{2} + ⋯ < ∞

for every ξ ∈ H.

It is apparent from the preceding remarks that the positive operator defined by the sum ∑_{k} v_{k}v_{k}^{∗} does not depend on the choice of orthonormal basis (v_{k}), and in fact we can associate with E a unique normal completely positive linear map P_{E} on M by

P_{E}(x) = ∑_{k} v_{k}xv_{k}^{∗}, x ∈ M.

The preceding remarks can now be summarized as follows: In the von Neumann algebra M = B(H), the association E ↔ P_{E} defines a bijective correspondence between the set of metric operator spaces contained in M and the set of normal completely positive linear maps in L(M).

Remark 1.13. Every normal completely positive linear map can be decomposed into a sum of the form P_{E_{0}} + c · ι_{M}, where c is a nonnegative scalar, ι_{M}4 is the identity map of M, and E_{0} is a metric operator space satisfying the condition E_{0} ∩ ℂ1 = {0}. To see that, suppose that E contains the identity operator and we set E_{0} = {v ∈ E : ⟨v, 1⟩_{E} = 0}. Then E_{0} is a metric operator space satisfying E_{0} ∩ ℂ1 = {0}, and notice that there is a positive scalar c such that

P_{E}(x) = P_{E_{0}}(x) + cx.

Indeed, we can choose an orthonormal basis v_{0}, v_{1}, v_{2}, … for E so that v_{0} = λ1 is a multiple of 1. Then v_{1}, v_{2}, … is an orthonormal basis for E_{0} and we have

P_{E}(x) = ∑_{k=0}^{∞} v_{k}xv_{k}^{∗} = ∑_{k=1}^{∞} v_{k}xv_{k}^{∗} + v_{0}xv_{0}^{∗} = P_{E_{0}}(x) + |λ|^{2}x.

Remark. While a normal completely positive map P on M determines its metric operator space E uniquely, that is not the case for the symbol of P. More precisely, if E_{1} and E_{2} are two metric operator spaces with respective completely positive maps P_{1} and P_{2}, then σ_{P_{1}} = σ_{P_{2}} iff E_{1} + ℂ1 = E_{2} + ℂ1. We will not make use of that fact, but we do require the following more explicit result from which it is easily deduced (see Theorem 3.3 and Remark 3.18 below for general results related to this issue).

Theorem 1.14. Let E be a metric operator space satisfying E ∩ ℂ · 1 = {0}, and let P = P_{E} be its completely positive map. The most general normal completely positive linear map Q ∈ L(M) satisfying σ_{Q} = σ_{P} has the form

Q(x) = ∑_{k}(v_{k} + λ_{k}1)x(v_{k} + λ_{k}1)^{∗} + cx,

where v_{1}, v_{2}, … is an orthonormal basis for E, (λ_{1}, λ_{2}, …) belongs to ℓ^{2}, and c is a nonnegative scalar.

proof. In view of (Remark 1.13), it suffices to prove the following assertion. Let E, Ẽ be metric operator spaces such that E ∩ ℂ1 = Ẽ ∩ ℂ1 = {0}, for which P_{E} and PẼ have the same symbol. Then there is an orthonormal basis v_{1}, v_{2}, … for E and an ℓ^{2} sequence λ_{1}, λ_{2}, … such that v_{1} + λ_{1}1, v_{2} + λ_{2}1, … is an orthonormal basis for Ẽ.

To prove this, choose an orthonormal basis u_{1}, u_{2}, … for E and let n = dim E. If we let π be the diagonal representation of M = B(H) on H^{n}

π(x) = x ⊕ x ⊕ …,

then we have

P_{E}(x) = ∑_{k} u_{k}xu_{k}^{∗} = V^{∗}π(x)V,

where V ∈ B(H, H^{n}) is the operator

V ξ = (u_{1}^{∗}ξ, u_{2}^{∗}ξ, …).

Because of (1.9) we have H^{n} = [π(x)ξ : x ∈ M, ξ ∈ H]. However in this case, since E ∩ ℂ1 = {0}, we claim that in fact

(1.15)  H^{n} = [(V x − π(x)V)ξ : x ∈ M, ξ ∈ H].

To prove that let K be the subspace of H^{n} defined by the right side of (1.15). Since the map x ∈ M ↦ D(x) = V x − π(x)V ∈ B(H, K) satisfies

D(xy) = π(x)D(y) + D(x)y,

it follows that K = [D(M)H] is invariant under operators in π(M), and hence the projection p onto the orthocomplement of K belongs to the commutant of π(M) and satisfies pD(x) = 0 for every x ∈ M. We have to show that p = 0. Considering the form of π we find that p is a matrix of scalar operators p = (λ_{ij}1). Each row of the scalar matrix (λ_{ij}) belongs to ℓ^{2}, and the condition pD(x) = 0 implies that for every i = 1, 2, … and every x ∈ M we have

∑_{j} λ_{ij}(u_{j}^{∗}x − xu_{j}^{∗}) = 0.

Thus for every i the operator w_{i} = ∑_{j} λ̄_{ij}u_{j} is an element of E which commutes with every operator in M, and is therefore a scalar multiple of the identity. Since E ∩ ℂ1 = {0} we conclude that w_{1} = w_{2} = ⋯ = 0. Hence by (1.9) the matrix p = (λ_{ij}1) is zero, proving (1.15).

Now let m = dim Ẽ and let

π̃(x) = x ⊕ x ⊕ …

be the corresponding representation of M on H^{m}. Let ũ_{1}, ũ_{2}, … be an orthonormal basis for Ẽ and let Ṽ : H → H^{m} be the associated operator

Ṽ ξ = (ũ_{1}^{∗}ξ, ũ_{2}^{∗}ξ, …).

Then there is a corresponding derivation D̃ : M → B(H, K) which is defined by D̃(x) = Ṽ x − π̃(x)Ṽ. We claim next that there is a (necessarily unique) unitary operator W : H^{n} → H^{m} satisfying

(1.16)  W D(x) = D̃(x),  x ∈ M.

Noting that H^{n} = [D(M)H] and H^{m} = [D̃(M)H], it is clearly enough to show that for all ξ, η ∈ H we have

(1.17)  ⟨D(x)ξ, D(y)η⟩ = ⟨D̃(x)ξ, D̃(y)η⟩ .

For that, we write

D(y)^{∗}D(x) = (V y − π(y)V)^{∗}(V x − π(x)V)
= V^{∗}π(xy)V − y^{∗}V^{∗}π(x)V − V^{∗}π(y^{∗})V x + y^{∗}V^{∗}V x
= σ_{P_{E}} (dy^{∗} dx).

By hypothesis the symbols of P_{E} and PẼ agree, hence the right side is

σẼ(dy^{∗} dx) = D̃(y)^{∗}D̃(x),

and formula (1.17) follows.

Note that W π(x) = π̃(x)W for every x. Indeed, fixing x and choosing a vector in H^{n} of the form η = D(y)ξ for y ∈ M, ξ ∈ H we have

W π(x)η = W π(x)D(y)ξ = W D(xy)ξ − W D(x)yξ
= D̃(xy)ξ − D̃(x)yξ = π̃(x)D̃(y)ξ = π̃(x)W D(y)ξ.

The assertion follows because H^{n} is spanned by such vectors η.

In particular, π and π̃ are equivalent representations of M. Hence m = n and therefore H^{m} = H^{n}. Moreover, W belongs to the commutant of π and hence there is a unitary matrix (λ_{ij}) of complex scalars such that W = (λ_{ij}1). If we now look at the components of the operator equation W D(x) = D̃(x) we find that for every i = 1, 2, …

∑_{j} λ_{ij}(u_{j}^{∗}x − xu_{j}^{∗}) = ũ_{i}^{∗}x − xũ_{i}^{∗}.

Thus we can define a new orthonormal basis v_{1}, v_{2}, … for E by

v_{i} = ∑_{j} λ̄_{ij}u_{j}.

The preceding equation relating the u_{j} to the ũ_{j} becomes

v_{i}^{∗}x − xv_{i}^{∗} = ũ_{i}^{∗}x − xũ_{i}^{∗}

for every operator x ∈ M = B(H). It follows that the operators ũ_{i} − v_{i} commute with all bounded operators and therefore must be scalar multiples of the identity operator, say

(1.18)  ũ_{i} = v_{i} + λ_{i}1.

The fact that both series ∑_{j} v_{j}v_{j}^{∗} and ∑ ũ_{j}ũ_{j}^{∗} converge strongly implies that ∑_{j} |λ|^{2} < ∞, and thus (1.18) provides an orthonormal basis ũ_{1}, ũ_{2}, … for Ẽ of the required form. □

Corollary. Let E and P = P_{E} satisfy the hypotheses of Theorem 1.14, and let k be an operator in M. The following are equivalent:

(1) Q(x) = P (x) + kx + xk^{∗} is completely positive.

(2) k has a (necessarily unique) decompostion of the form k = v + c1, where v ∈ E and c is a complex number satisfying c + c̄ ≥ ⟨v, v⟩_{E}. proof of (2) ⟹ (1). Let k = v + 1/2 ⟨v, v⟩_{E} + d1 where v ∈ E and d is a complex number with nonnegative real part. Choose an orthonormal basis v_{1}, v_{2}, … for E and set λ_{k} = ⟨v, v_{k}⟩. Then for α = d + d̄− ⟨v, v⟩_{E} ≥ 0 we have

kx + xk^{∗} = vx + xv^{∗} + (⟨v, v⟩_{E} + α)x = ∑_{k}(λ_{k}v_{k}x + λ̄_{k}xv_{k}^{∗} + |λ_{k}|^{2}x) + αx,

hence the map Q of (1) can be written

Q(x) = ∑_{k}(v_{k} + λ̄_{k}1)x(v_{k} + λ̄_{k}1)^{∗} + αx

which is obviously completely positive.

Before proving the opposite implication we collect two elementary observations.

Lemma 1.19.

(1) The only completely positive linear map of M = B(H) having symbol 0 is of the form L(x) = cx where c is a nonnegative scalar.

(2) The only operator k ∈ M for which P (x) = kx+xk^{∗} is a completely positive map is of the form k = z1, where z is a complex number having nonnegative real part.

proof. Suppose that P is a completely positive linear map on M for which σ_{P} = 0. Let P (x) = V^{∗}π(x)V be a Stinespring representation for P. From the definition of σ_{P} one finds that

σ_{P} (dx dy) = (V^{∗}π(x) − xV^{∗})(π(y)V − V y).

Therefore (π(x)V − V x)^{∗}(π(x)V − V x) = σ_{P} (d(x^{∗}) dx) = 0, hence V x = π(x)V for every x. It follows that V^{∗}V x = xV^{∗}V for all x ∈ M = B(H) so that there is a nonnegative scalar x such that V^{∗}V = c1. Thus P (x) = V^{∗}π(x)V = V^{∗}V x = cx as asserted.

For the second assertion, suppose that x ↦ kx + xk^{∗} is completely positive. The symbol of this map vanishes, so by what was just proved there is a nonnegative scalar c such that kx+xk^{∗} = cx for every x. Taking x to be an arbitrary projection p we find that (1 − p)kp = 0, hence k must be a scalar k = λ1. The formula kx + xk^{∗} = cx implies that λ + λ̄ = c ≥ 0, and the Lemma is proved. □

proof of (1) ⟹ (2). Suppose that k is an operator in M such that Q(x) = P (x) + kx + xk^{∗} is completely positive. Then Q and P have the same symbol, hence by Theorem 1.14 there is an ℓ^{2} sequence λ = (λ_{1}, λ_{2}, …) and an orthonormal basis v_{1}, v_{2}, … for E such that

Q(x) = ∑_{k}(v_{k} + λ_{k}1)x(v_{k} + λ_{k}1)^{∗}.

Define an element v ∈ E by v = ∑_{k} λ̄_{k}v_{k}, and write |v|^{2} = ⟨v, v⟩_{E} = ∑_{k} |λ_{k}|^{2}. We find that

Q(x) = ∑_{k} v_{k}xv_{k}^{∗} + vx + xv^{∗} + |v|^{2}x
= P (x) + (v + 1/2|v|^{2}1)x + x(v + 1/2|v|^{2}1)^{∗}
= P (x) + kx + xk^{∗}.

It follows that the operator ℓ = k −v −1/2|v|^{2}1 has the property that x ↦ ℓx+xℓ^{∗} is a completely positive linear map on M = B(H). From part (2) of the preceding Lemma we find that there is a complex scalar z having nonnegative real part such that ℓ = z1, and the required representation

k = v + (1/2|v|^{2} + z)1

follows □

We conclude this section by reformulating a known description of the bounded generators of semigroups of completely positive maps in terms of metric operator spaces.

Proposition 1.20. Let L ∈ L(M) be an operator which generates a semigroup P_{t} = exp(tL), t ≥ 0 of normal completely positive maps on M. Then there is a metric operator space E satisfying E ∩ ℂ1 = {0}, and an operator z ∈ M such that

L(x) = P_{E}(x) + zx + xz^{∗} x ∈ M.

proof. Using a general result of Christensen and Evans [7], one can find a completely positive linear map Q : M → M and an element z ∈ M such that

(1.21)  L(x) = Q(x) + zx + xz^{∗},  x ∈ M.

Since L is a bounded operator that generates a semigroup of normal completely positive maps, it must itself be a normal linear map on M; and we may conclude from (1.21) that Q is normal. Let E be a metric operator space such that Q = P_{E}. The remark following Definition 1.12 shows that we can arrange E ∩ ℂ1 = {0} by adjusting z if necessary □

Remarks. Unlike the case of completely positive maps, the correspondence between metric operator spaces and generators of CP semigroups is not quite one-to-one. However, if (E_{1}, z_{1}) and (E_{2}, z_{2}) are two pairs which serve to represent a given generator L as in Proposition 1.20 and which satisfy

(1.22)  E_{1} ∩ ℂ1 = E_{2} ∩ ℂ1 = {0},

then it follows from Theorem 1.14 that E_{1} + ℂ1 = E_{2} + ℂ1, and hence dim E_{1} = dim E_{2}. Thus we can make the following

Definition 1.23. Let L be a bounded operator on B(H) which generates a semigroup of normal completely positive maps on B(H). The rank of L is defined as the dimension of any metric operator space E for which E ∩ ℂ1 = {0} and which gives rise to a representation of L in the form

L(x) = P_{E}(x) + kx + xk^{∗}, x ∈ B(H)

where k is some operator in B(H).

We also point out that, with a little more care, one can recover the inner product on E from the properties of L (see Remark 3.18 below).

2. Units and covariance function.

In [4], a notion of index for CP semigroups P was introduced which directly generalizes the definition of numerical index of E_{0}-semigroups. Briefly, a unit of P is a strongly continuous semigroup T = {T (t) : t ≥ 0} of bounded operators in M for which there is a real constant k with the property that for every t ≥ 0, the operator mapping

x ∈ M ↦ e^{kt}P_{t}(x) − T (t)xT (t)^{∗}

is completely positive. Let U_{P} be the set of all units of P. It is possible for U_{P} to be the empty set; indeed there are E_{0}-semigroups with this property [9,10]. But if U_{P} ≠ ∅ then one can define a function

c_{P} : U_{P} × U_{P} → ℂ,

called the covariance function of the semigroup P, as follows. Note first that for every positive t, there is a unique metric operator space associated with the completely positive map P_{t}; we will write E_{P} (t) for this metric operator space. The most elementary properties of the family

E_{P} = {E_{P} (t) : t > 0}

are as follows: (2.1.1) Each E_{P} (t) is a separable Hilbert space. (2.1.2) E_{P} (s + t) is spanned as a Hilbert space by the set of products

{xy : x ∈ E_{P} (s), y ∈ E_{P} (t)}.

Remarks. Regarding (2.1.2), it is shown in [4, Theorem 1.12] that operator multiplication

u ⊗ v ∈ E_{P} (s) ⊗ E_{P} (t) ↦ uv ∈ E_{P} (s + t)

extends uniquely to a bounded linear operator from the Hilbert space E_{P} (s)⊗E_{P} (t) to E_{P} (s + t) whose adjoint is an isometry from E_{P} (s + t) onto a closed subspace of E_{P} (s) ⊗ E_{P} (t).

In order to define the covariance function, choose T_{1}, T_{2} ∈ U_{P}, and fix t > 0. The condition (2.1) implies that both operators T_{1}(t) and T_{2}(t) belong to E_{P} (t) and thus we can form their inner product ⟨T_{1}(t), T_{2}(t)⟩_{E_{P} (t)} as elements of this Hilbert space. More generally, if we are given an arbitrary finite partition

P = {0 = t_{0} < t_{1} < ⋯ < t_{n} = t}

of the interval [0, t] then we define a function f_{P,t} : U_{p} × U_{P} → ℂ as follows

f_{P,t}(T_{1}, T_{2}) = ∏_{k=1}^{n} ⟨T_{1}((t_{k} − t_{k−1}), T_{2}(t_{k} − t_{k−1})⟩_{E_{P} (t_{k}−t_{k−1})} .

It was shown in [4] that there is a unique complex number c_{P} (T_{1}, T_{2}) such that for every t > 0

(2.2)  lim_{P} f_{P,t}(T_{1}, T_{2}) = e^{tc_{P} (T_{1},T_{2})},

the limit being taken over the increasing directed set of all finite partitions P of [0, t]. That defines the covariance function c_{P} : U_{P} × U_{P} → ℂ of any CP semigroup P for which U_{P} ≠ ∅.

The covariance function is conditionally positive definite in the sense that if T_{1}, T_{2}, …, T_{n} ∈ U_{P} and λ_{1}, λ_{2}, …, λ_{n} ∈ ℂ satisfy λ_{1} + λ_{2} + ⋯ + λ_{n} = 0, then

∑_{i,j=1}^{n} λ_{i}λ̄_{j}c_{P} (T_{i}, T_{j}) ≥ 0

(see [cpindex, Proposition 2.7]). More generally, if we are given any nonempty set X and a conditionally positive definite function c : X × X → ℂ then there is a natural way to construct a Hilbert space H(X, c). Briefly, c defines a positive semidefinite sesquilinear form ⟨·, ·⟩ on the vector space V of all finitely nonzero functions f : X → ℂ for which

∑_{x∈X} f(x) = 0,

by way of

⟨f, g⟩ = ∑_{x,y∈X} f(x)ḡ(y)c(x, y),

and H(X, c) is obtained by completing the inner product space obtained by promoting ⟨·, ·⟩ to the quotient of V by the subspace

N = {f ∈ V : ⟨f, f⟩ = 0}.

The index d_{∗}(P) of a CP semigroup P is defined by

d_{∗}(P) = dim H(U_{P} , c_{P})

in the case where U_{P} ≠ ∅, and is defined by d_{∗}(P) = 2^{ℵ_{0}} if U_{P} = ∅. Notice that in order to calculate d_{∗}(P) one must calculate a) the set U_{P} of all units of P, and b) the covariance function c_{P} : U_{P} × U_{P} → ℂ; moreover, this must be done explicitly enough so that the dimension of H(U_{P} , c_{P}) is apparent. The purpose of this section is to carry out these calculations for the case of CP semigroups with bounded generator in terms of the structures associated with the generator by Proposition 1.20. The principal result is Theorem 2.3 below.

Remarks. The covariance function c_{P} : U_{P} × U_{P} → ℂ is a conditionally positive definite function having the property that for every t > 0,

T_{1}, T_{2} ∈ U_{P} ↦ e^{tc_{P} (T_{1},T_{2})} − ⟨T_{1}(t), T_{2}(t)⟩_{E_{P} (t)}

is a positive definite function. It is a simple exercise to show that if d : U_{P} ×U_{P} → ℂ is any other function for which

T_{1}, T_{2} ∈ U_{P} ↦ e^{td(T_{1},T_{2})} − ⟨T_{1}(t), T_{2}(t)⟩_{E_{P} (t)}

is positive definite for every t > 0, then the difference d − c_{P} is a positive definite function on U_{P} × U_{P}. Thus the covariance function c_{P} is characterized in this sense as the “smallest” function c : U_{P} × U_{P} → ℂ with the property that e^{tc(T_{1},T_{2})} dominates the inner products ⟨T_{1}(t), T_{2}(t)⟩_{E_{P} (t)} for every t > 0.

Theorem 2.3. Let L ∈ L(M) be a bounded operator which generates a CP semigroup on M = B(H). Let E be a metric operator space satisfying the conditions of Proposition 1.20, so that L has the form

(2.3.1)  L(x) = P_{E}(x) + kx + xk^{∗}

for some k ∈ M. The units of the CP semigroup P = {exp(tL) : t ≥ 0} are described in terms of E and k as follows. For every (c, v) ∈ ℂ × E, let T_{(c,v)} be the operator semigroup

(2.3.2)  T_{(c,v)}(t) = e^{ct} exp t(v + k),  t ≥ 0.

Then T_{(c,v)} is a unit of P and the map (c, v) ∈ ℂ × E ↦ T_{(c,v)} is a bijection of ℂ × E onto the set U_{P} of units of P.

The covariance function c_{P} : U_{P} × U_{P} ↦ ℂ of P is given by

(2.3.2)  c_{P} (T_{(c_{1},v_{1})}, T_{(c_{2},v_{2})}) = c_{1} + c̄_{2} + ⟨v_{1}, v_{2}⟩_{E} ,

and the index of P is d_{∗}(P) = dim E.

proof. Notice first that the map (c, v) → T_{(c,v)} from ℂ × E to operator semigroups is one-to-one. Indeed, choosing complex numbers c_{1}, c_{2} and elements v_{1}, v_{2} ∈ E such that T_{(c_{1},v_{1})}(t) = T_{(c_{2},v_{2})}(t) for every t, it follows that the generators of these two semigroups are equal. Using (2.3.2) we find that

c_{1}1 + v_{1} + k = c_{2}1 + v_{2} + k.

Cancelling k and using E ∩ ℂ1 = {0}, we obtain v_{1} = v_{2} and c_{1} = c_{2} as asserted.

Now fix (c, v) ∈ ℂ × E. We will show that T_{(c,v)} is a unit of P. In order to do that, we must find a real constant α such that each mapping

(2.4)  x ∈ M ↦ e^{tα}P_{t}(x) − T_{(c,v)}(t)xT_{(c,v)}(t)^{∗}

is completely positive, t ≥ 0. Noting that T_{(c,v)}(t) = e^{ct}T_{(0,v)}(t), it is clearly enough to prove (2.4) for c = 0; in that case we will show that (2.4) is true for the constant α = ⟨v, v⟩_{E}.

To that end, we set

L_{0}(x) = (v + k)x + x(v + k)^{∗},

and we claim first that for α = ⟨v, v⟩_{E} we have

(2.5)  L_{0} ≤ L + α · ι_{M},

ι_{M} denoting the identity map of M. Equivalently, after choosing an orthonormal basis v_{1}, v_{2}, … for E, we want to show that the mapping

(2.6)  x ↦ ∑_{j} v_{j}xv_{j}^{∗} − vx − xv^{∗} + αx

is completely positive. Indeed, since v ∈ E and (v_{j}) is an orthonormal basis, we can find a sequence λ ∈ ℓ^{2} such that v = ∑_{j} λ̄_{j}v_{j} (note the complex conjugate). Then

α = ⟨v, v⟩ = ∑_{j} |λ_{j}|^{2},

and hence the term on the right side of (2.6) can be collected as follows,

∑_{j}(v_{j} − λ_{j}1)x(v_{j} − λ_{j})^{∗}.

The latter formula obviously defines a completely positive map.

In order to pass from (2.5) to its exponentiated version (2.4), we require Lemma 2.7. Let L_{1}, L_{2} belong to L(M). Suppose that both generate CP semigroups and that L_{2} − L_{1} is completely positive. Then for every t ≥ 0 the map

exp(tL_{2}) − exp(tL_{1})

is completely positive.

proof. Since the hypotheses on L_{1} and L_{2} are invariant under scaling by positive constants, it is enough to prove the assertion for t = 1. We can write L_{2} = L_{1} + R where R is a completely positive map. By the Lie product formula [12, page 245], we have

exp(L_{2}) = exp(L_{1} + R) = lim_{n→∞}(exp(1/nL_{1}) exp(1/nR))^{n},

the limit on n existing relative to the operator norm on L(M). Thus it suffices to show that for every n, we have

(2.8)  (exp(1/nL_{1}) exp(1/nR))^{n} ≥ exp L_{1}.

To see the latter, note that for completely positive maps A_{k}, B_{k}, k = 1, 2 we have

B_{1} ≥ A_{1} and B_{2} ≥ A_{2} ⟹ B_{1}B_{2} ≥ A_{1}A_{2}.

Indeed, this follows from the fact that a composition of completely positive maps is completely positive, so that A_{1}(B_{2} − A_{2}) ≥ 0 and (B_{1} − A_{1})B_{2} ≥ 0 together imply that B_{1}B_{2} ≥ A_{1}B_{2} ≥ A_{1}A_{2}, and the assertion follows.

We apply this to (2.8) as follows. Letting ι_{M} denote the identity map of M we have

exp(1/nR) = ι_{M} + 1/nR + 1/2(1/nR)^{2} + ⋯ ≥ ι_{M}

because R is completely positive. Hence

exp(1/nL_{1}) exp(1/nR) ≥ exp(1/nL_{1}).

For the same reason,

(exp(1/nL_{1}) exp(1/nR))^{2} ≥ (exp(1/nL_{1}))^{2},

and so on until we obtain

(exp(1/nL_{1}) exp(1/nR))^{n} ≥ (exp(1/nL_{1}))^{n} = exp(L_{1}).

This establishes (2.8) and completes the proof of Lemma 2.7 □

Applying Lemma 2.7 to (2.5) we obtain

exp(tL_{0}) ≤ exp(t(L + αι_{M})) = e^{tα}P_{t}.

Noting that L_{0}(x) = (v + k)x + x(v + k)^{∗} is the generator of the semigroup

x ↦ T_{(0,v)}(t)xT_{(0,v)}(t)^{∗},

we obtain formula (2.4) for the case c = 0. This completes the proof that each operator semigroup of the form T_{(c,v)} ∈ U_{P} is a unit of P.

We show next that the map (c, v) ∈ ℝ × E ↦ T_{(c,v)} ∈ U_{P}. is surjective. For that, the following result is essential.

Lemma 2.9. Let L be a bounded operator in L(M) which generates a unital CP semigroup. Then every semigroup T ∈ U_{P} has a bounded generator.

proof. Let T be an operator semigroup with the property that for every t ≥ 0 the mapping

(2.10)  x ↦ e^{αt}P_{t}(x) − T (t)xT (t)^{∗}

is completely positive, P_{t} denotine exp(tL) and α being some real constant. To show that the generator of T is a bounded operator, it is enough to show that T is continuous relative to the operator norm on M

(2.11)  lim_{t→0+} ∥T (t) − 1∥ = 0.

To prove (2.11) we make use of the symbol as follows. Fix t > 0 and consider the operator mapping

L_{t}(x) = T (t)xT (t)^{∗}, x ∈ M.

The symbol of L_{t} is found to be

σ_{L_{t}}(dx dy) = (T (t)x − xT (t))(T (t)^{∗}y − yT (t)^{∗}).

Now the symbol of a completely positive map R satisfies σ_{R}(dx dx^{∗}) ≥ 0. Hence if R_{1} and R_{2} are completely positive maps satisfying 0 ≤ R_{1} ≤ R_{2} then we have 0 ≤ σ_{L_{1}}(dx dx^{∗}) ≤ σ_{L_{2}}(dx dx^{∗}). Thus, (2,10) implies that for all x ∈ M,

(T (t)x − xT (t))(T (t)x − xT (t))^{∗} = σ_{L_{t}}(dx dx^{∗}) ≤ e^{tα}σ_{P_{t}}(dx dx^{∗})
= e^{tα}σ_{P_{t}−ι_{M}} (dx dx^{∗}),

the last equality resulting from the fact that the identity map ι_{M} of M has symbol zero. From (1.4) and the previous formula we conclude that for all x ∈ M satisfying ∥x∥ ≤ 1, we have

∥T (t)x − xT (t)∥^{2} = e^{tα}∥σ_{P_{t}−ι_{M}} (dx dx^{∗})∥ ≤ 4e^{tα}∥P_{t} − ι_{M}∥,

∥P_{t} − ι_{M}∥ denoting the norm of P_{t} − ι_{M} as an element of L(M). Now

∥P_{t} − ι_{M}∥ = ∥ exp(tL) − ι_{M}∥ → 0

as t → 0 because L is bounded. Since the norm of a derivation of M = B(H) of the form D(x) = T x − xT satisfies inequalities of the form

inf_{λ∈ℂ} ∥T − λ1∥ ≤ ∥D∥ ≤ 2 inf_{λ∈ℂ} ∥T − λ1∥,

it follows that

inf_{λ∈ℂ} ∥T (t) − λ1∥ ≤ sup_{∥x∥≤1} ∥T (t)x − xT (t)∥ → 0,

as t → 0. Thus there exist complex scalars λ_{t} such that ∥T (t) − λ_{t}1∥ → 0 as t → 0. Since the semigroup {T (t) : t ≥ 0} is strongly continuous, T (t) must tend to 1 in the strong operator topology as t → 0; hence λ_{t} → 1 as t → 0 and (2.11) follows □

Now choose any unit T ∈ U_{P}. There is a real constant α such that for every t the mapping

x ↦ e^{tα}P_{t}(x) − T (t)xT (t)^{∗}

is completely positive. We will show that there is an element (c, v) ∈ ℂ × E such that T = T_{(c,v)}. By replacing T with the semigroup {e^{−αt/2}T (t) : t ≥ 0} (and adjusting c accordingly), we may clearly assume that α = 0. By Lemma 2.9, there is a bounded operator a ∈ M such that

T (t) = exp(ta), t ≥ 0

and we have to show that a has the form

(2.12)  a = c1 + v + k

for some complex scalar c and some v ∈ E. For that, we claim first that the operator mapping

(2.13)  R(x) = L(x) − ax − xa^{∗}

is completely positive. Indeed, since for every t > 0 the map

x ↦ P_{t}(x) − e^{ta}xe^{ta^{∗}} = (P_{t}(x) − x) − (e^{ta}xe^{ta^{∗}} − x)

is completely positive, we may divide the latter by t and take the limit as t → 0+ to obtain (2.13), after noting that

lim_{t→0+} t^{−1}(P_{t}(x) − x) = L(x),  and
lim_{t→0+} t^{−1}(e^{ta}xe^{ta^{∗}} − x) = ax + xa^{∗}.

Using (2.3.1) we can write R in the form

R(x) = P_{E}(x) + (k − a)x + x(k − a)^{∗}.

The Corollary of Theorem 1.14 implies that there is a complex number d and an element u ∈ E such that

k − a = d1 + u,

and the required representation (2.12) follows after taking c = −d and v = −u.

It remains to compute the covariance function c_{P} of formula (2.2) in these coordinates.

Lemma 2.14. Let v_{1}, …, v_{n} ∈ E and let T_{j} be the unit of P defined by

T_{j}(t) = exp t(v_{j} + k), t ≥ 0, 1 ≤ j ≤ n.

Then for every λ_{1}, …, λ_{n} ∈ ℂ we have

∑_{i,j} λ_{i}λ̄_{j} ⟨v_{i}, v_{j}⟩_{E} ≤ ∑_{i,j} λ_{i}λ̄_{j}c_{P} (T_{i}, T_{j}).

proof. Notice that if an n-tuple λ_{1}, …, λ_{n} satisfies the required inequality and c is an arbitrary complex number, then so does cλ_{1}, …, cλ_{n}. Thus it is enough to prove Lemma 2.14 for n-tuples λ_{k} satisfying λ_{1} + λ_{2} + ⋯ + λ_{n} = 1.

Choose T_{1}, …, T_{n} ∈ U_{P}. By the remarks following (2.2) we have

∑_{i,j} λ_{i}λ̄_{j} ⟨T_{i}(t), T_{j}(t)⟩_{E_{P} (t)} ≤ ∑_{i,j} λ_{i}λ̄_{j}e^{tc_{P} (T_{i},T_{j})}.

Equivalently, if for every t > 0 we set

A(t) = λ_{1}T_{1}(t) + ⋯ + λ_{n}T_{n}(t),

then A(t) ∈ E_{P} (t) and we have

⟨A(t), A(t)⟩_{E_{P} (t)} ≤ ∑_{i,j} λ_{i}λ̄_{j}e^{tc_{P} (T_{i},T_{j})}.

It follows that the mapping

x ↦ (∑_{i,j} λ_{i}λ̄_{j}e^{tc_{P} (T_{i},T_{j})})P_{t}(x) − A(t)xA(t)^{∗}

is completely positive. Since ∑_{j} λ_{j} = 1, this implies that

x ↦ ∑_{i,j} λ_{i}λ̄_{j}(e^{tc_{P} (T_{i},T_{j})}P_{t}(x) − x) − (A(t)xA(t)^{∗} − x)

is completely positive. Notice that A(t) is differentiable at t = 0+ and that A′(0+) = ∑_{j} λ_{j}v_{j} + k. Thus if we divide by t and take lim_{t→0+} we find that

x ↦ (L(x) + ∑_{i,j} λ_{i}λ̄_{j}c_{P} (T_{i}, T_{j})) − ((∑_{j} λ_{j}v_{j})x + x(∑_{j} λ_{j}v_{j})^{∗} + kx + xk^{∗})

is a completely positive map. Noting that L(x) = P_{E}(x) + kx + xk^{∗}, the terms involving k and k^{∗} cancel and we are left with a completely positive map of the form

x ↦ P_{E}(x) − v(λ)x − xv(λ)^{∗} + (∑_{i,j} λ_{i}λ̄_{j}c_{P} (T_{i}, T_{j}))x.

where v(λ) = ∑_{j} λ_{j}v_{j} ∈ E. From the corollary of Theorem 1.14 we deduce the required inequality

∑_{i,j} λ_{i}λ̄_{j}c_{P} (T_{i}, T_{j}) ≥ ⟨v(λ), v(λ)⟩_{E} = ∑_{i,j} λ_{i}λ̄_{j} ⟨v_{i}, v_{j}⟩_{E} .

□ Lemma 2.15. Let v ∈ E and let T ∈ U_{P} be the semigroup T (t) = exp t(v + k), t ≥ 0. Then

c_{P} (T, T) ≤ ⟨v, v⟩_{E} .

proof. It has already been shown (see (2.4)) that the mapping

x ↦ e^{t⟨v,v⟩_{E}} P_{t}(x) − T (t)xT (t)^{∗}

is completely positive. It follows from the definition of E_{P} (t) that

⟨T (t), T (t)⟩_{E_{P} (t)} ≤ e^{t⟨v,v⟩_{E}}

for every t > 0. Thus for every finite partition

P = {0 = t_{0} < t_{1} < ⋯ < t_{n} = t}

of the interval [0, t] we have

∏_{k=1}^{n} ⟨T (t_{k} − t_{k−1}), T (t_{k} − t_{k−1})⟩_{E_{P} (t_{k}−t_{k−1})} ≤ e^{t⟨v,v⟩_{E}} .

Noting the definition (2.2) of c_{P} we conclude that

e^{tc_{P} (T,T)} ≤ e^{t⟨v,v⟩_{E}}

for every t > 0, and the asserted inequality follows □

To complete the proof of Theorem 2.3, choose complex numbers c_{1}, c_{2}, choose v_{1}, v_{2} ∈ E, and let T_{1}, T_{2} be the units of P defined by

T_{k}(t) = T_{(0,v_{k})}e^{ct} exp t(v_{k} + k), t ≥ 0.

Consider the self-adjoint 2 × 2 matrices A = (a_{ij}) and B = (b_{ij}) defined by

a_{ij} = ⟨v_{i}, v_{j}⟩_{E} ,
b_{ij} = c_{P} (T_{i}, T_{j}).

Lemma 2.14 implies that B − A ≥ 0, while Lemma 2.15 implies that the diagonal terms of B − A are nonpositive. Hence the trace of B − A is nonpositive and it follows that A = B. Comparing the off-diagonal terms we obtain

(2.16)  c_{P} (T_{1}, T_{2}) = ⟨v_{1}, v_{2}⟩_{E} .

Now T_{k} = T_{(0,v_{k})}. We bring in the c_{k} as follows. From the definition (2.2) of c_{P} and the fact that T_{(c,v)}(t) = e^{ct}T_{(0,v)}(t), it follows that

c_{P} (T_{(c_{1},v_{1})}, T_{(c_{2},v_{2})}) = c_{1} + c̄_{2} + c_{P} (T_{(0,v_{1})}, T_{(0,v_{2})}).

Together with (2.16), this implies

c_{P} (T_{(c_{1},v_{1})}, T_{(c_{2},v_{2})}) = c_{1} + c̄_{2} + ⟨v_{1}, v_{2}⟩_{E} ,

as required.

Once one has this formula we obtain d_{∗}(P) = dim E by a straightforward calculation. Indeed, the preceding arguments show that we may identify U_{P} with ℂ × E in such a way that the covariance function becomes

c_{P} ((a, v), (b, w)) = a + b̄ + ⟨v, w⟩ .

Now as in [1, Proposition 5.3] one computes directly that

d_{∗}(P) = dim H(ℂ × E, c_{P}) = dim E.

□

Together with the results of [4], Theorem 2.3 shows how to calculate the index of the minimal E_{0}-semigroup dilation of any unital CP semigroup having bounded generator:

Corollary 2.17. Let L be a bounded linear map on B(H) which generates a semigroup P = {P_{t} = exp(tL) : t ≥ 0} of normal completely positive maps satisfying P_{t}(1) = 1 for every t ≥ 0. Let α = {α_{t} : t ≥ 0} be the minimal E_{0}-semigroup-dilation of P. Then U_{α} ≠ ∅ and the numerical index d_{∗}(α) is the rank of L.

proof. By Proposition 1.20, we find an operator k and a metric operator space E satisfying E ∩ ℂ1 = {0}, and

L(x) = P_{E}(x) + kx + xk^{∗}, x ∈ B(H).

By [cpindex, Theorem 4.9] we have d_{∗}(α) = d_{∗}(P), while by Theorem 2.3 above we have d_{∗}(p) = dim E = rank(L). □

Remarks. It is possible, of course, for the rank of L to be 0; equivalently, E = {0} and hence ℂ × E ≅ ℂ. In this event P is a semigroup of ∗-automorphisms of B(H) and α = P. This degenerate case is discussed more fully in the proof of Corollary 4.25 below.

3. Completeness of the covariance function. It is possible for different CP semigroups P, Q to have the same covariance function in the sense that P and Q have the same set of units and

c_{P} (S, T) = c_{Q}(S, T), S, T ∈ U_{P} = U_{Q}.

For example, let α = {α_{t} : t ≥ 0} be an E_{0}-semigroup acting on M = B(H). For every t > 0 let E_{α}(t) be the metric operator space associated with α_{t}. Since α_{t} is an endomorphism we have in this case

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

and the inner product in E_{α}(t) is defined by

⟨S, T ⟩_{E_{α}(t)} 1 = T^{∗}S.

Assuming that U_{α} ≠ ∅, we can form a closed subspace D(t) of the Hilbert space E_{α}(t) generated by all finite products obtained from units as follows

D(t) = [u_{1}(t_{1})u_{2}(t_{2}) … u_{n}(t_{n}) : u_{k} ∈ U_{α}, t_{k} > 0, t_{1} + ⋯ + t_{n} = t, n ≥ 1}.

D(t) is itself a metric operator space, and it determines a ∗-endomorphism β_{t} of B(H) by way of

β_{t}(x) = ∑_{k} v_{k}xv_{k}^{∗}, x ∈ B(H)

{v_{1}, v_{2} … } denoting an orthonormal basis for D(t). Since D(s + t) is spanned by the set of products {ST : S ∈ D(s), T ∈ D(t)} it follows that β_{s+t} = β_{s}β_{t}. If we set β_{0} = ι_{B(H)}, then β = {β_{t} : t ≥ 0} is a semigroup of normal ∗-endomorphisms of B(H). The individual maps β_{t} are not necessarily unit preserving, but we do have

(3.2)  β_{t} ≤ α_{t},  t ≥ 0,

and in fact the semigroup β is continuous. Moreover, the following conditions are equivalent for every t > 0,

CS1 H is spanned by {T ξ : T ∈ D(t), ξ ∈ H} CS2 β_{t}(1) = 1 CS3 E_{α}(t) = D(t) CS4 β_{t} = α_{t}

(see [1, §7]), and when these conditions are satisfied for some t > 0 then they are satisfied for every t > 0. When that is the case, α is called completely spatial. If α is any semigroup for which U_{α} ≠ ∅ then α is called spatial and in this case we refer to its associated semigroup β as the standard part of α.

Now a straightforward computation shows that α and β have the same set of units and the same covariance function. So if α is any spatial E_{0}-semigroup which is not completely spatial, then α, β provide rather extreme examples of distinct CP semigroups which have the same covariance function. The existence of such E_{0}- semigroups is established in [10]. The following result asserts that this phenomenon cannot occur for CP semigroups which have bounded generators.

Theorem 3.3. Let P_{1}, P_{2} be CP semigroups with bounded generators L_{1}, L_{2}. Suppose that P_{1} and P_{2} have the same set of units and

c_{P_{1}}(T, T′) = c_{P_{2}}(T, T′), T, T′ ∈ U_{P_{1}} = U_{P_{2}}.

Then L_{1} = L_{2}, and hence P_{1} = P_{2}.

poroof. By Proposition 1.20 we can find metric operator spaces E_{1}, E_{2} and operators k_{1}, k_{2} ∈ B(H) satisfying

(3.4)  E_{1} ∩ ℂ1 = E_{2} ∩ ℂ1 = {0}

and

L_{j}(x) = P_{E_{j}}(x) + k_{j}x + xk_{j}^{∗}, x ∈ B(H)

for j = 1, 2. Theorem 2.3 asserts that the most general unit of P_{j} is a semigroup of the form T (t) = exp(ta), where a belongs to the set of operators E_{j} + ℂ1 + k_{j}. Thus, the hypothesis U_{P_{1}} = U_{P_{2}} is equivalent to the equality of the two sets

E_{1} + ℂ1 + k_{1} = E_{2} + ℂ1 + k_{2}.

Now if E_{1} and E_{2} are linear subspaces of a vector space V and k_{1}, k_{2} are elements of V such that E_{1} + k_{1} = E_{2} + k_{2} then we must have E_{1} = E_{2} and k_{2} − k_{1} ∈ E_{2}. Taking E_{j} = E_{j} + ℂ1 it follows that

(3.6)  E_{1} + ℂ1 = E_{2} + ℂ1,

and

(3.7)  k_{2} = k_{1} + v_{2} + λ1,

where v_{2} ∈ E_{2} and λ is a scalar.

Associated with any pair of operator spaces E_{1}, E_{2} satisfying (3.4) and (3.6) there is an isomorphism of vector spaces θ : E_{1} → E_{2}. Indeed, since every element v ∈ E_{1} has a unique decomposition

v = v′ + λ1

where v′ ∈ E_{2} and λ ∈ ℂ, we can define a linear functional f : E_{1} → ℂ and a linear isomorphism θ : E_{1} → E_{2} by

(3.8)  v = θ(v) + f(v)1.

We will show first that the linear isomorphism θ : E_{1} → E_{2} defined by (3.8) is a unitary operator in that for any pair of elements v, v′ ∈ E_{1} we have ⟨θ(v), θ(v′)⟩_{E_{2}} = ⟨v, v′⟩_{E_{1}}. To that end, fix v, v′ ∈ E_{1} and consider the units T, T′ of P_{1} defined by

T (t) = exp t(v + k_{1}), T′(t) = exp t(v′ + k_{1}).

Combining formula (2.2) with Theorem 2.3, we find that for every t > 0,

(3.9)  c_{P_{1}}(T, T′) = ⟨v, v′⟩_{E_{1}} .

Now we must consider T and T′ relative to the coordinates associated with P_{2}. In order to do that, we use (3.7) and (3.8) to write

v + k_{1} = θ(v) + f(v)1 + k_{1} = (θ(v) − v_{2}) + (f(v) − λ)1 + k_{2},

and similarly

v′ + k_{1} = (θ(v′) − v_{2}) + (f(v′) − λ)1 + k_{2}.

Considering T and T′ as units of P_{2}, we have in the notation of formula (2.3.3),

T = T_{(f(v)−λ,θ(v)−v_{2})}, T′ = T_{(f(v′)−λ,θ(v′)−v_{2})},

and corresponding to (3.9) we have

(3.10)  c_{P_{2}}(T, T′) = f(v) − λ + f̄(v′) − λ̄ + ⟨θ(v) − v_{2}, θ(v′) − v_{2}⟩_{E_{2}} .

Since c_{P_{1}} = c_{P_{2}}, we may equate the right sides of (2.23) and (2.24) to obtain

(3.11)  ⟨v, v′⟩_{E_{1}} = f(v) − λ + f̄(v′) − λ̄ + ⟨θ(v) − v_{2}, θ(v′) − v_{2}⟩_{E_{2}} .

The identity (3.11) implies that θ is unitary. To see that, consider the sesquilinear form representing the difference

D(v, v′) = ⟨v, v′⟩_{E_{1}} − ⟨θ(v), θ(v′)⟩_{E_{2}} .

We can rewrite (2.25) in the form

(3.12)  D(v, v′) = g(v) + ḡ(v′),

where g : E_{1} → ℂ is the function g(v) = f(v) − ⟨θ(v), v_{2}⟩_{E_{1}} + 1/2 ⟨v_{2}, v_{2}⟩_{E_{2}} − λ. For every t > 0 we can write

D(v, v′) = t^{−2}D(tv, tv′) = t^{−2}(g(tv) + ḡ(tv′)),

and clearly

t^{−2}g(tv) = t^{−1}(f(v) − ⟨θ(v), v_{2}⟩) + t^{−2}(1/2 ⟨v_{2}, v_{2}⟩_{E_{2}} − λ)

tends to zero as t → ∞. Thus,

(3.13)  D(v, v′) = 0 = g(v) + ḡ(v′)

for every v, v′ ∈ E_{1}.

We claim next that (3.13) implies

(3.14)  λ = 1/2 ⟨v_{2}, v_{2}⟩_{E_{2}} + ic,  and
(3.15)  f(v) = ⟨θ(v), v_{2}⟩_{E_{2}} ,

where i = \sqrt{−1} and c is a real number. Indeed, setting v = v′ = 0 in the equation g(v) + ḡ(v′) = 0 (3.13) we obtain

1/2 ⟨v_{2}, v_{2}⟩_{E_{2}} − λ + 1/2 ⟨v_{2}, v_{2}⟩_{E_{2}} − λ̄ = 0,

hence (3.14). Thus the linear functional ρ(v) = f(v) − ⟨θ(v), v_{2}⟩_{E_{2}} satisfies

ρ(v) + ρ̄(v′) = g(v) + ḡ(v′) = 0

for all v, v′ ∈ E_{1} and (3.15) follows after setting v′ = 0.

From (3.7) and (3.14) we obtain

k_{2} = k_{1} + v_{2} + 1/2 ⟨v_{2}, v_{2}⟩_{E_{2}} 1 + ic1,

so for all x ∈ B(H) we have

(3.16)  k_{2}x + xk_{2}^{∗} = k_{1}x + xk_{1}^{∗} + v_{2}x + xv_{2}^{∗} + ⟨v_{2}, v_{2}⟩_{E_{2}} x.

Since L_{2}(x) = P_{E_{2}}(x) + k_{2}x + xk_{2}^{∗} it follows that

(3.17)  L_{2}(x) = P_{E_{2}}(x) + v_{2}x + xv_{2}^{∗} + ⟨v_{2}, v_{2}⟩_{E_{2}} x + k_{1}x + xk_{1}^{∗}.

We will show now that the right side of (3.17) is L_{1}(x); equivalently, we claim that

(3.18)  P_{E_{2}}(x) + v_{2}x + xv_{2}^{∗} + ⟨v_{2}, v_{2}⟩_{E_{2}} x = P_{E_{1}}(x).

To see that, let u_{1}, u_{2}, … be an orthonormal basis for E_{1}. Then θ(u_{1}), θ(u_{2}) … is an orthonormal basis for E_{2} and if we set μ_{k} = ⟨v_{2}, θ(u_{k})⟩_{E_{2}} then the sequence (μ_{1}, μ_{2}, …) belongs to ℓ^{2} and v_{2} = ∑_{k} μ_{k}θ(u_{k}). Thus we have

v_{2}x + xv_{2}^{∗} + ⟨v_{2}, v_{2}⟩_{E_{2}} x = ∑_{k}(μ_{k}θ(u_{k})x + xμ̄_{k}θ(u_{k})^{∗} + |μ_{k}|^{2}x),

while

P_{E_{2}}(x) = ∑_{k} θ(u_{k})xθ(u_{k})^{∗},

so that the left side of (3.18) can be written

∑_{k}(θ(u_{k}) + μ̄_{k}1)x(θ(u_{k}) + μ̄_{k}1)^{∗}.

Noting that for each k,

θ(u_{k}) + μ̄_{k}1 = θ(u_{k}) + ⟨θ(u_{k}), v_{2}⟩_{E_{2}} 1 = θ(u_{k}) + f(u_{k})1 = u_{k}

by the definition (3.8) of θ and f, we find that the last expression reduces to

∑_{k} u_{k}xu_{k}^{∗} = P_{E_{1}}(x),

as asserted. That completes the proof of Theorem 3.3 □

Remark 3.18. The proof of Theorem 3.3 gives somewhat more information than is contained in its statement. For example, suppose that E_{1}, E_{2} are two metric operator spaces satisfying E_{1} ∩ ℂ1 = E_{2} ∩ 1 = {0}, and let k_{1}, k_{2} be two operators such that the corresponding generators are the same:

P_{E_{1}}(x) + k_{1}x + xk_{1}^{∗} = P_{E_{2}} + k_{2}x + xk_{2}^{∗}, x ∈ B(H).

Then the proof of Theorem 3.3 implies that

(3.19)  E_{1} + ℂ1 = E_{2} + ℂ1.

(3.19) allows one to define a linear isomorphism θ : E_{1} → E_{2} and a linear functional f on E_{1} by

(3.20)  v = θ(v) + f(v)1,  v ∈ E_{1}.

The same argument shows that θ is a unitary operator and that the unique element v_{2} ∈ E_{2} defined by

(3.21)  f(v) = ⟨θ(v), v_{2}⟩_{E_{2}} ,  v ∈ E_{1}

satisfies

(3.22)  k_{2} = k_{1} + v_{2} + (1/2 ⟨v_{2}, v_{2}⟩_{E_{2}} + ic)1

where c is a real constant.

Conversely, if we start with two pairs (E_{1}, k_{1}), (E_{2}, k_{2}) satisfying (3.19)–(3.22) (so that the linear isomorphism θ defined by (3.20) is a unitary operator) along with E_{1} ∩ ℂ1 = E_{2} ∩ ℂ1 = {0}, then both (E_{1}, k_{1}) and (E_{2}, k_{2}) give rise to the same generator

L(x) = P_{E_{1}}(x) + k_{1}x + xk_{1}^{∗} = P_{E_{2}}(x) + k_{2}x + xk_{2}^{∗}.

4. Minimal dilations.

Every unital CP semigroup P = {P_{t} : t ≥ 0} acting on B(H) has a minimal dilation to an E_{0}-semigroup α = {α_{t} : t ≥ 0} acting on B(K), where K is a Hilbert space which contains H as a subspace [2,5,6]. Recall that an E_{0}-semigroup α is said to be completely spatial if there are “enough” units in the sense that the equivalent conditions CS1–CS4 of the preceding section are satisfied. The completely spatial E_{0}-semigroups constitute the best-understood class. All of the basic examples are of this type, and they are classified up to cocycle conjugacy by their numerical index d_{∗}(α) [1, Corollary of Proposition 7.2]. Thus, if one knows that an E_{0}-semigroup α is completely spatial and has numerical index n = 1, 2, …, ∞, then α must be conjugate to a cocycle perturbation of the CAR/CCR flow of index n.

The purpose of this section is to show that if a CP semigroup has a bounded generator then its minimal dilation is completely spatial (Theorem 4.8 below). The proof of Theorem 4.8 is based on Theorem 3.3 and the following result.

Theorem 4.1. Let P = {P_{t} : t ≥ 0} be a CP semigroup acting on B(H) which has a bounded generator. Let {Q_{t} : t > 0} be a family of normal completely positive maps on B(H) satisfying the two conditions

(4.1.1)  Q_{t} ≤ P_{t},
(4.1.2)  Q_{s+t} = Q_{s}Q_{t},

for all s, t > 0, and which is not the trivial family Q_{t} = 0, t > 0. Let Q_{0} be the identity map of B(H). Then {Q_{t} : t ≥ 0} is also a CP semigroup having bounded generator.

Our proof of this result requires the following estimate.

Lemma 4.2. Let P be a normal completely positive linear map on M = B(H) and let σ_{P} be its symbol. Then we have

inf_{λ>0} ∥P − λι_{M}∥ ≤ 6∥P ∥^{1/2} sup_{∥x∥≤1} ∥σ_{P} (dx^{∗} dx)∥^{1/2},

where ι_{M} denotes the identity map of M.

Remark. In proving this estimate we will make use of the following bit of lore. Let N ⊆ B(K) be an amenable von Neumann algebra and let T be an operator on K. Then there is an operator T′ in the commutant of N such that

(4.3)  ∥T − T′∥ = sup{∥T x − xT ∥ : x ∈ N, ∥x∥ ≤ 1}.

Indeed, the operator T′ may be obtained by a familiar averaging process, in which one uses an invariant mean on a suitable subgroup G of the unitary group in N to average quantities of the form uT u^{∗}, u ∈ G, after noting that for every such unitary operator u, ∥uT u^{∗} − T ∥ is dominated by the right side of (4.3).

proof of Lemma 4.2. Because of Stinespring’s theorem there is a Hilbert space K, a normal representation π : M → B(K) and an operator V ∈ B(H, K) such that P (x) = V^{∗}π(x)V, x ∈ M. As in the proof of Lemma 1.19, the symbol of P is related to V and π by way of

(π(x)V − V x)^{∗}(π(x)V − V x) = σ_{P} (dx^{∗} dx),

and hence

∥V x − π(x)V ∥^{2} = ∥σ_{P} (dx^{∗} dx)∥.

Setting ∥σ_{P} ∥ = sup_{∥x∥≤1} ∥σ_{P} (dx^{∗} dx)∥, we obtain

(4.4)  sup_{∥x∥≤1} ∥V x − π(x)V ∥ ≤ ∥σ_{P} ∥^{1/2}.

Now consider the von Neumann algebra N ⊆ B(H ⊕ K) and the operator T ∈ B(H ⊕ K) defined by

N = \{( \begin{matrix}x & 0 \\ 0 & π(x) \end{matrix} ) : x ∈ M\}, T = ( \begin{matrix}0 & V^{∗} \\ V & 0 \end{matrix} ) .

One finds that

T ( \begin{matrix}x & 0 \\ 0 & π(x) \end{matrix} ) − ( \begin{matrix}x & 0 \\ 0 & π(x) \end{matrix} ) T = ( \begin{matrix}0 & −(V x − π(x)V)^{∗} \\ V x − π(x)V & 0 \end{matrix} ) .

The norm of the operator on the right is ∥V x − π(x)V ∥, hence

sup{∥T y − yT ∥ : y ∈ N, ∥y∥ ≤ 1} =
sup{∥V x − π(x)V ∥ : x ∈ M, ∥x∥ ≤ 1} ≤ ∥σ_{P} ∥^{1/2}.

Using (4.3) we find an operator T′ ∈ N′ satisfying ∥T − T′∥ ≤ ∥σ_{P} ∥^{1/2}. A straightforward computation shows that operators in the commutant of N must have the form

T′ = ( \begin{matrix}A & Y^{∗} \\ X & B \end{matrix} )

where A is a scalar multiple of the identity of B(H), B belongs to the commutant of π(M), and X and Y are intertwining operators, Xx = π(x)X, Y x = π(x)Y, x ∈ M. It follows that there is such an X for which

(4.5)  ∥V − X∥ ≤ ∥T − T′∥ ≤ ∥σ_{P} ∥^{1/2}.

Since X^{∗}X commutes with M = B(H) we must have X^{∗}X = λ1_{H} for some scalar λ ≥ 0, and hence

∥P (x) − λ · x∥ = ∥V^{∗}π(x)V − X^{∗}π(x)X∥ ≤ 2∥V − X∥ · ∥x∥ max (∥V ∥, ∥X∥).

Note that max (∥V ∥, ∥X∥) ≤ 3∥P ∥^{1/2}. Indeed, since V^{∗}V = P (1) we have ∥V ∥ ≤ ∥P ∥^{1/2}, and by (1.4) we can estimate ∥X∥ by way of

∥X∥ ≤ ∥V ∥ + ∥V − X∥ ≤ ∥P ∥^{1/2} + ∥σ_{P} ∥^{1/2} ≤ 3∥P ∥^{1/2}.

Using (4.5) we arrive at the desired inequality ∥P − λ · ι_{M}∥ ≤ 6∥P ∥^{1/2}∥σ_{P} ∥^{1/2} □

proof of Theorem 4.1. Let {Q_{t} : t > 0} satisfy (4.1.1) and (4.1.2). We will show that

(4.6)  lim_{t→0+} ∥Q_{t} − ι∥ = 0,

ι denoting the identity map ι(x) = x, x ∈ B(H). Theorem 4.1 follows immediately since under these conditions the semigroup {Q_{t} : t ≥ 0} becomes a continuous semigroup of elements in the Banach algebra of all normal linear mappings L on B(H) with the uniform norm

∥L∥ = sup_{∥x∥≤1} ∥L(x)∥.

In order to prove (4.6), we claim first that there is a family λ_{t}, t > 0 of nonnegative numbers such that ∥Q_{t} − λ_{t}ι∥ → 0 as t → 0+. Indeed, since P_{t} − Q_{t} is completely positive for every t > 0 we have 0 ≤ σ_{Q_{t}}(dx dx^{∗}) ≤ σ_{P_{t}}(dx dx^{∗}) for every x ∈ M; hence

∥σ_{Q_{t}}(dx dx^{∗})∥ ≤ ∥σ_{P_{t}}(dx dx^{∗})∥ = ∥σ_{P_{t}−ι}(dx dx^{∗})∥ ≤ 4∥P_{t} − ι∥ ∥x∥^{2}.

Using Lemma 4.2 together with the latter inequality we find that

inf_{λ>0} ∥Q_{t} − λι∥ ≤ 6∥Q_{t}∥^{1/2} sup_{∥x∥≤1} ∥σ_{Q_{t}}(dx dx^{∗})∥^{1/2} ≤
12∥Q_{t}∥^{1/2}∥P_{t} − ι∥^{1/2} ≤ 12∥P_{t}∥^{1/2}∥P_{t} − ι∥^{1/2}

and the claim follows because ∥P_{t} − ι∥ tends to 0 as t → 0+.

It remains to prove that λ_{t} → 1 as t → 0+. To that end, we claim

(4.7)  lim_{t→0+} ∥Q_{t}∥ = 1.

Indeed, since ∥Q_{t}∥ ≤ ∥P_{t}∥ and ∥P_{t} − ι∥ → 0 as t → 0+, we have

lim sup_{t→0+} ∥Q_{t}∥ ≤ lim sup_{t→0+} ∥P_{t}∥ = ∥ι∥ = 1.

So if (4.7) fails then we must have

lim inf_{t→0+} ∥Q_{t}∥ < 1,

and in that event we can pick r < 1 such that lim inf_{s→0+} ∥Q_{s}∥ < r. Let R > 1 be close enough to 1 so that rR < 1. Then for sufficiently small t we have both ∥Q_{t}∥ ≤ R and inf_{0<s<t} ∥Q_{s}∥ ≤ r. For such a t we can find 0 < s < t such that ∥Q_{s}∥ ≤ r, and hence

∥Q_{t}∥ ≤ ∥Q_{s}∥ · ∥Q_{t−s}∥ ≤ rR.

Thus ∥Q_{t}∥ ≤ ∥Q_{t/n}∥^{n} ≤ (rR)^{n} for every n = 1, 2, … and hence ∥Q_{t}∥ = 0 for all sufficiently small t. Because of the semigroup property it follows that ∥Q_{t}∥ is identically zero, contradicting our hypothesis on Q and proving (4.7).

To see that λ_{t} → 1 as t → 0+ write

λ_{t} = ∥(λ_{t}ι − Q_{t}) + Q_{t}∥,

and use lim_{t→0+} ∥Q_{t} − λ_{t}ι∥ = 0 together with (4.7) □

Following is the principal result of this section.

Theorem 4.8. Let P = {P_{t} : t ≥ 0} be a unital CP semigroup having a bounded generator. Then the minimal dilation of P to an E_{0}-semigroup is completely spatial.

proof. Let α = {α_{t} : t ≥ 0} be an E_{0}-semigroup acting on B(K), K ⊇ H which is the minimal dilation of P. Let β = {β_{t} : t ≥ 0} be the standard part of α as defined in section 3. Letting {D(t) : t > 0} be the family of metric operator spaces of (3.1), we have to show that

K = [Aξ : A ∈ D(t), ξ ∈ K],

or equivalently, that β_{t}(1_{K}) = 1_{K} for every t.

Now in general, since β is a semigroup of endomorphisms of B(K), the projections β_{t}(1_{K}) decrease as t increases, and the limit projection

(4.9)  e_{∞} = lim_{t→∞} β_{t}(1_{K})

is fixed under the action of β. The compression β^{∞} of the semigroup β to the corner e_{∞}B(K)e_{∞} can be viewed as a semigroup of unit-preserving endomorphisms of B(e_{∞}K). It is not quite obvious that β^{∞} is an E_{0}-semigroup since we have not proved that β is continuous. While it is possible to establish that directly, we will not have to do so since the following result implies that β^{∞} is actually a compression of α.

Lemma 4.10. The projection e_{∞} satisfies α_{t}(e_{∞}) ≥ e_{∞} for every t ≥ 0, and the compression of α_{t} to e_{∞}B(K)e_{∞} is β_{t}^{∞} for every t ≥ 0. In particular, β^{∞} defines an E_{0}-semigroup acting on B(e_{∞}K).

proof. We claim first that for every x ∈ B(K) we have

(4.11)  α_{t}(x)β_{t}(1_{K}) = β_{t}(x) = β_{t}(1_{K})α_{t}(x).

Indeed, if we let u_{1}, u_{2}, … be an orthonormal basis for D(t) then we have

β_{t}(x) = ∑_{j} u_{j}xu_{j}^{∗}, x ∈ B(K).

Since D(t) ⊆ E_{α}(t) it follows that for every j,

α_{t}(x)u_{j}u_{j}^{∗} = u_{j}xu_{j}^{∗} = u_{j}u_{j}^{∗}α_{t}(x),

and (4.11) follows by summing on j. Taking x = e_{∞} in (4.11) and using e_{∞} = β_{t}(e_{∞})e_{∞}, we obtain

α_{t}(e_{∞})e_{∞} = α_{t}(e_{∞})β_{t}(e_{∞})e_{∞} = β_{t}(e_{∞})e_{∞} = e_{∞},

hence α_{t}(e_{∞}) ≥ e_{∞}.

Now choose an operator x ∈ e_{∞}B(K)e_{∞}. We have

e_{∞}α_{t}(x)e_{∞} = e_{∞}β_{t}(1_{K})α_{t}(x)e_{∞} = e_{∞}β_{t}(x)e_{∞}
= β_{t}(e_{∞})β_{t}(x)β_{t}(e_{∞}) = β_{t}(e_{∞}xe_{∞}) = β_{t}(x),

as asserted □

Let p_{0} ∈ B(K) be the projection onto H. Then of course α_{t}(p_{0}) ≥ p_{0}, and after identifying B(H) with p_{0}B(K)p_{0} we have

(4.12)  P_{t}(x) = p_{0}α_{t}(x)p_{0},  t ≥ 0, x ∈ B(H).

We will prove that

(4.13)  p_{0} ≤ e_{∞} = lim_{t→∞} β_{t}(1_{K}).

Granting (4.13) for a moment, then Lemma 4.10 asserts that the compression of α to e_{∞}B(K)e_{∞} is an E_{0}-semigroup which is obviously an intermediate dilation of P. By the definition of minimal dilation given in [2] we may conclude that e_{∞} = 1_{K}. In particular, β_{t}(1_{K}) = 1_{K} for every t, proving that α is completely spatial.

Thus the proof of Theorem 4.8 is reduced to establishing (4.13). This will be done indirectly, by considering the family of maps Q = {Q_{t} : t ≥ 0} on B(H) obtained by compressing the semigroup β:

(4.14)  Q_{t}(x) = p_{0}β_{t}(x)p_{0},  x ∈ B(H), t ≥ 0.

We will show that Q is a CP semigroup with bounded generator, which has the same set of units and the same covariance function as P. Theorem 3.3 will then imply that Q_{t} = P_{t} for all t, and in particular Q_{t}(p_{0}) = P_{t}(P_{0}) = p_{0}. From the definition (4.14) of Q we conclude that

p_{0} ≤ β_{t}(p_{0}) ≤ β_{t}(1_{K})

for every t ≥ 0, and (4.13) will follow.

We show first that the family Q of (4.14) is a semigroup. To that end, we claim that for every t ≥ 0,

(4.15)  p_{0}β_{t}(p_{0}) = p_{0}β_{t}(1_{K}), and β_{t}(p_{0})p_{0} = β_{t}(1_{K})p_{0}.

Indeed, since β_{t}(x) = α_{t}(x)β_{t}(1_{K}) for every x ∈ B(K) we have

p_{0}β_{t}(1_{K} − p_{0}) = p_{0}α_{t}(1_{K} − p_{0})β_{t}(1_{K}) = p_{0}(1_{K} − α_{t}(p_{0}))β_{t}(1_{K})

and the right side is zero because α_{t}(p_{0}) ≥ p_{0}. Similarly, β_{t}(1_{K} − p_{0})p_{0} = 0. Thus for s, t > 0 and x ∈ B(H) we have

Q_{s}Q_{t}(x) = p_{0}β_{s}(p_{0}β_{t}(x)p_{0})p_{0} = p_{0}β_{s}(p_{0})β_{s+t}(x)β_{s}(p_{0})p_{0}.

Because of (4.15) we have p_{0}β_{s}(p_{0}) = p_{0}β_{s}(1_{K}) and β_{s}(p_{0})p_{0} = β_{s}(1_{K})p_{0}, hence

Q_{s}Q_{t}(x) = p_{0}β_{s}(1_{K})β_{s+t}(x)β_{s}(1_{K})p_{0}.

Finally, since β_{s+t}(1_{K}) ≤ β_{s}(1_{K}), we have

β_{s}(1_{K})β_{s+t}(x)β_{s}(1_{K}) = β_{s+t}(x),

and the formula Q_{s}Q_{t}(x) = Q_{s+t}(x) follows.

Now since β_{t} ≤ α_{t} for every t ≥ 0 we have Q_{t} ≤ P_{t}. Thus the hypotheses (4.1.1) and (4.1.2) of Theorem 4.1 are satisfied. Notice that Q_{t} cannot vanish for every t > 0. Indeed, Corollary 2.17 implies that α must have units. Choose u ∈ U_{α}, and let T be the corresponding unit of P defined by

T (t)^{∗} = u(t)^{∗} ↾_{H}, t ≥ 0.

There is a real constant k such that ⟨u(t), u(t)⟩_{E_{α}(t)} = e^{kt}. Since u(t) ∈ D(t) we have

u(t)u(t)^{∗} ≤ e^{kt}β_{t}(1_{K}),

and hence by (4.15)

T (t)T (t)^{∗} = p_{0}u(t)u(t)^{∗}p_{0} ≤ e^{kt}p_{0}β_{t}(1_{K})p_{0} = e^{kt}p_{0}β_{t}(p_{0})p_{0} = Q_{t}(p_{0}).

Since T (t)T (t)^{∗} tends weakly to p_{0} = 1_{H} as t → 0+, so does Q_{t}(p_{0}).

From Theorem 4.1 we conclude that the semigroup Q has a bounded generator. Finally, we claim that Q_{t} = P_{t} for all t ≥ 0. According to Theorem 3.3, it is enough to show that U_{P} = U_{Q} and that P and Q have the same covariance function. For that, we require a general observation:

Lemma 4.16. Let P, Q be two CP semigroups acting on B(H), and suppose that Q_{t} ≤ P_{t} for every t ≥ 0. Then U_{Q} ⊆ U_{P}, and for every finite set T_{1}, T_{2}, …, T_{n} ∈ U_{Q} and λ_{1}, λ_{2}, …, λ_{n} ∈ ℂ we have

∑_{i,j} λ_{i}λ̄_{j}c_{P} (T_{i}, T_{j}) ≤ ∑_{i,j} λ_{i}λ̄_{j}c_{Q}(T_{i}, T_{j}),

where c_{P} and c_{Q} are the covariance functions of P and Q.

Remarks. Let X be a set and f : X × X → ℂ a function. Recall that f is positive definite if for every x_{1}, x_{2}, …, x_{n} ∈ X and λ_{1}, λ_{2}, …, λ_{n} ∈ ℂ we have

∑_{i,j} λ_{i}λ̄_{j}f(x_{i}, x_{j}) ≥ 0.

Given two functions f, g : X × X → ℂ, we will write f ≲ g or g ≳ f if g − f is positive definite. We will make use of the following elementary facts about positive definite functions.

(4.17)  f ≳ 0, g ≳ 0 ⟹ fg ≳ 0,

where fg denotes the pointwise product fg(x, y) = f(x, y)g(x, y). (4.13), together with transitivity of the relation ≲, implies that for any four complex-valued functions f_{1}, f_{2}, g_{2}, g_{2} on X × X we have

(4.18)  0 ≲ f_{i} ≲ g_{i}, i = 1, 2 ⟹ f_{1}f_{2} ≲ g_{1}g_{2}.

proof of Lemma 4.16. Choose T ∈ U_{Q}. Then there is a real constant k such that the semigroup R_{t}(x) = T (t)xT (t)^{∗} satisfies

R_{t} ≤ e^{kt}Q_{t} ≤ e^{kt}P_{t}, t ≥ 0,

hence T ∈ U_{P}.

Fix S, T ∈ U_{Q}, and for every t > 0 define functions f, g by

f(S, T; t) = ⟨S(t), T (t)⟩_{E_{Q}(t)} ,
g(S, T; t) = ⟨S(t), T (t)⟩_{E_{P} (t)} .

Notice first that

(4.19)  0 ≲ f(·, ·; t) ≲ g(·, ·; t)

as functions on U_{Q} × U_{Q}. Indeed, if T_{1}, T_{2}, …, T_{n} ∈ U_{Q}, λ_{1}, λ_{2}, …, λ_{n} ∈ ℂ and we set A(t) = ∑_{k} λ_{k}T_{k}(t), then A(t) belongs to E_{Q}(t) and hence the mapping x ↦ A(t)xA(t)^{∗} is dominated by

⟨A(t), A(t)⟩_{E_{Q}(t)} Q_{t} ≤ ⟨A(t), A(t)⟩_{E_{Q}(t)} P_{t}.

It follows that

0 ≤ ⟨A(t), A(t)⟩_{E_{P} (t)} ≤ ⟨A(t), A(t)⟩_{E_{Q}(t)}

and (4.19) follows after expanding the inner products in the obvious way. For every partition P = {0 = t_{0} < t_{1}, ⋯ < t_{n} = t} of the interval [0, t], set

f_{P}(S, T; t) = ∏_{k−1}^{n} ⟨S(t_{k} − t_{k−1}), T (t_{k} − t_{k−1})⟩_{E_{P} (t)} ,

g_{P}(S, T; t) = ∏_{k−1}^{n} ⟨S(t_{k} − t_{k−1}), T (t_{k} − t_{k−1})⟩_{E_{Q}(t)} .

By (4.17) and (4.18), we have

0 ≲ f_{P}(·, ·; t) ≲ g_{P}(·, ·; t)

for every partition P of [0, t]. After taking the limit on P we obtain

e^{tc_{P}} ≲ e^{tc_{Q}},

for every t > 0. It follows that e^{tc_{P}} − 1 ≲ e^{tc_{Q}} − 1 for every t > 0 and hence

c_{P} = lim_{t→0+} \frac{e^{tc_{P}} − 1}{t} ≲ lim_{t→0+} \frac{e^{tc_{Q}} − 1}{t} = c_{Q},

as required □

We claim now that

(4.19)  U_{P} ⊆ U_{Q},  and
(4.20)  c_{Q}(T, T) ≤ c_{P} (T, T),  for every T ∈ U_{P} .

To see that, choose any unit T ∈ U_{P}. By [4, Theorem 3.6] there is a unit u ∈ U_{α} such that

T (t)^{∗} = u(t)^{∗} ↾_{H}, t > 0;

moreover, c_{P} (T, T) = c_{α}(u, u) because of the minimality of α. Since u(t) ∈ D(t) and ⟨u(t), u(t)⟩_{D(t)} = e^{tc_{α}(u,u)}, the map

x ∈ B(H) ↦ e^{tc_{α}(u,u)}Q_{t}(x) − T (t)xT (t)^{∗}

is completely positive. (4.19) follows. We may also conclude from this argument that

⟨T (t), T (t)⟩_{E_{Q}(t)} ≤ e^{tc_{P} (T,T)}.

Hence for every partition P = {0 = t_{0} < t_{1} < ⋯ < t_{n} = t} of the interval [0, t],

∏_{k=1}^{n} ⟨T (t_{k} − t_{k−1}), T (t_{k} − t_{k−1})⟩_{E_{Q}(t)} ≤ e^{tc_{P} (T,T)},

and after passing to the limit on P we obtain

e^{tc_{Q}(T,T)} ≤ e^{tc_{P} (T,T)}

for every t > 0, from which (4.20) is immediate.

Together with Lemma 4.12, (4.19) implies that U_{P} = U_{Q}. We claim now that c_{P} = c_{Q}. To see that, fix T_{1}, T_{2} ∈ U_{P}, and consider the 2 × 2 matrix A = (a_{ij}) defined by

a_{ij} = c_{Q}(T_{i}, T_{j}) − c_{P} (T_{i}, T_{j}).

Lemma 4.16 implies that A ≥ 0; while (4.20) implies that both diagonal terms of A are nonpositive, so that the trace of A is nonpositive. It follows that A = 0. In particular,

c_{Q}(T_{1}, T_{2}) − c_{P} (T_{1}, t_{2}) = a_{ij} = 0.

We may now apply Theorem 3.3 to obtain P = Q. As we have already pointed out, one may deduce from this the required inequality (4.13). That completes the proof of Theorem 4.8 □

Corollary 4.21. Let P = {P_{t} : t ≥ 0} be a unital CP semigroup acting on B(H) whose generator is bounded and which is not a semigroup of ∗-automorphisms. Then the minimal dilation of P is a cocycle perturbation of a CAR/CCR flow of positive index r = 1, 2, …, ∞. The index r is the rank of the generator of P.

proof. By Theorem 4.8, the minimal dilation of P is a completely spatial E_{0}- semigroup. The classification results of [1, Corollary of Proposition 7.2] imply that this E_{0}-semigroup is cocycle conjugate to a CAR/CCR flow. Its index is the rank of the generator of P by Corollary 2.17.

Thus we only have to check that the generator cannot have rank zero. Let L be the generator of P and suppose that L has rank zero. Then the metric operator space associated with L is {0}, and L must have the form

L(x) = kx + xk^{∗}

for some k ∈ M_{n}(ℂ) (see Definition 1.23). Since P is unital we have L(1) = 0, hence k + k^{∗} = 0. It follows that k = ih for some self adjoint matrix h and L(x) = (ih)x − x(ih). Thus

P_{t}(x) = e^{ith}xe^{−ith}

is a semigroup of ∗-automorphisms, contrary to hypothesis □

Remarks. In particular, Corollary 4.21 leads to a description of the minimal dilations of all unital CP semigroups which act on a matrix algebra M_{n}(ℂ), n = 2, 3, . . .. If the semigroup is nontrivial then its minimal dilation α is a cocycle perturbation of a CAR/CCR flow of finite positive index d_{∗}(α). Considering the relation between generators and metric operator spaces (Proposition 1.20), we find that for fixed n the possible values of d_{∗}(α) are 1, 2, …, n^{2} − 1.

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