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To appear in Bulletin of the London Mathematical Society
EXISTENCE OF FELLER COCYCLES ON A C ∗ -ALGEBRA
J MARTIN LINDSAY AND STEPHEN J WILLS
Abstract. The quantum stochastic differential equation dkt = kt ◦ θβα dΛβ
α (t)
is considered on a unital C ∗ -algebra, with separable noise dimension space.
Necessary conditions on the matrix of bounded linear maps θ for the existence
of a completely positive contractive solution are shown to be sufficient. It is
known that for completely positive contraction processes, k satisfies such an
equation if and only if k is a regular Markovian cocycle. Feller refers to an
invariance condition analogous to probabilistic terminology if the algebra is
thought of as a noncommutative topological space.
0. Introduction
The quantum stochastic differential equation
dkt = kt ◦ θβα dΛβα (t),
k0 (a) = a ⊗ 1,
(0.1)
was introduced by Evans and Hudson ([EvH]) as a noncommutative generalisation of the SDE’s which govern stochastic flows of diffeomorphisms on a manifold
([Kun]). Noncommutativity appears both in the geometry ([Con]) and the noise
([Par]), since the equation is driven by linear maps acting on an operator algebra, and the differentials of the equation are quantum stochastic processes on a
symmetric Fock space.
The data for such equations is a triple (A, η, θ) consisting of a ∗ -algebra A of
bounded operators on a Hilbert space h, called the initial space, an orthonormal
basis η = (ei )i≥1 for a separable Hilbert space k, called the noise dimension space,
and a matrix θ = [θβα ]α,β≥0 of linear maps on A. Viewing k as a subspace of
b
k = C⊕k, the basis η is extended to a basis ηb = (eα )α≥0 of b
k by putting e0 = 1. With
respect to this basis, θ is the coefficient matrix of a (densely defined) sesquilinear
map Θ : b
k×b
k → L(A) and for us Λ = [Λα
β ]α,β≥0 is the quantum noise matrix of
Boson Fock stochastic calculus
"
# "
#
Λ00 (t) Λ0j (t)
t1
a∗ (ej 1[0,t] )
α
θβ = Θ(eα , eβ );
=
,
Λi0 (t) Λij (t)
a(ei 1[0,t] ) dΓ(|ej ihei | ⊗ M[0,t] )
a∗ , dΓ and a being respectively Fock space creation, differential second quantisation
and annihilation, and M[0,t] being the multiplication operator by the indicator
function 1[0,t] on L2 (R+ ) ([Par]). In most of the work so far on this equation each
θβα is assumed to belong to B(A) and, when k is infinite dimensional, A is assumed
to be a von Neumann algebra and θβα is assumed to be normal. In the current
work the former assumption stands, but the latter does not: for us A is a unital
C ∗ -algebra.
The existence of a solution of (0.1) was established by Evans, for finite dimensional k, using Picard iteration ([Eva]). The estimates used there were refined by
Mohari and Sinha who extended the results to infinite dimensional noise space, and
matrices θ for which each column θβ := [θβα ]α≥0 defines an operator A → B(h; h⊗ b
k)
which satisfies the relative boundedness conditions kθβ (a)uk ≤ k(a ⊗ 1K )Bβ uk, for
1991 Mathematics Subject Classification. 81S25, 46L07, 46L53, 47D06.
1
2
J MARTIN LINDSAY AND STEPHEN J WILLS
some supplementary Hilbert space K and bounded operator Bβ ([MoS]). In Meyer’s
exposition of this result ([Mey]) the validity of the existence theorem is extended
to matrices θ which satisfy the growth conditions
sup{(n!)−1 γ n Cnθ (a, u, d)2 : n ≥ 1} < ∞
for γ > 0, a ∈ A, u ∈ h and d ≥ 0, where
X X
Cnθ (a, u, d)2 =
kθβαnn ◦ · · · ◦ θβα11 (a)uk2 .
(0.2)
(0.3)
0≤βi ≤d αi ≥0
All of this work focused on the construction of ∗ -homomorphic solutions to (0.1),
and the characterisation of those θ which generate such a solution. Motivated by
quantum filtering theory, the search for a noncommutative analogue of the theory
of measure-valued diffusion, and the desire to see both flows on a C ∗ -algebra and
their Markov semigroups from a common viewpoint, attention has broadened to the
class of equations whose solutions are completely positive, first for finite dimensional
noise ([LiP]), and subsequently for infinite dimensional noise ([LW1]); see also [Bel].
The characterisation of those matrices θ that generate completely positive contraction flows obtained in [LW1] starts by assuming the existence of a (weakly regular,
weak) solution, and it is shown that θ must be completely bounded. When A is a
von Neumann algebra and θ is normal, if θ is also completely bounded then it necessarily satisfies the Mohari-Sinha regularity conditions ([LW1], Proposition 1.3),
and so the existence assumption holds. Here we address the C ∗ -situation.
Weak solutions of (0.1) (which are weakly regular) form Markovian cocycles
with respect to the right shift semigroup (σ̌s )s≥0 on the quantum noise ([LW2],
Proposition 5.2). When A is a von Neumann algebra and each kt is completely
bounded and normal, the cocycle identity reads simply
ks+t = b
ks ◦ σs ◦ kt , k0 (a) = a ⊗ 1
(0.4)
where b
ks = ks ⊗ idRan(σ̌s ) and σs = idA ⊗σ̌s ([Bra]). However, when A is just assumed to be a C ∗ -algebra the interpretation of (0.4) must be refined. The resulting
definition (see (2.4) below) includes invariance of the C ∗ -algebra under semigroups
associated with the process — this may be viewed as a Feller type condition if A
is thought of as a noncommutative topological space. In ([LW2]) all such Feller
cocycles on a unital C ∗ -algebra that are Markov regular, completely positive and
contractive are shown to strongly satisfy a QSDE of the form (0.1).
The purpose of this note is to prove that matrices θ whose columns are completely bounded necessarily satisfy (a strong form of) the regularity condition (0.2)
identified by Meyer, so that (0.1) has a strong solution. The above mentioned
characterisation of completely positive contraction flows is therefore freed of the
existence assumption, and thereby attains a more satisfactory form which moreover is manifestly basis independent.
Notations and conventions. Greek indices vary over the set 0 ≤ α < D whereas
roman indices vary over 1 ≤ i < D, where D = 1 + dim k; Einstein’s summation
convention for repeated indices is used throughout. Algebraic tensor products are
denoted . For a function f defined on R+ , and subset I of R+ , fI := f 1I where 1I
is the indicator function of I; L is used for spaces of linear maps, and B (respectively
CB) is used for spaces of bounded (resp. completely bounded) linear operators.
For a vector e in a Hilbert space h we use the notation Ee for each bounded
Hilbert space operator u 7→ u ⊗ e, and E e for its adjoint. Thus Ee ∈ B(H; H ⊗ h)
and E e may be thought of as a coordinate map. The Hilbert space H will always
be made clear by the context. Note that the linear maps e 7→ Ee are isometric
h → B(H; H ⊗ h). When an orthonormal basis (ei )i∈I for h is understood, E (i)
will denote E u for u = ei , and the tensor product H ⊗ h will be identified with the
EXISTENCE OF FELLER COCYCLES
3
L
(i)
orthogonal direct sum
i∈I H by the prescription ξ 7→ (E ξ)i∈I . The resulting
partial Parseval identity is useful
X
kξk2 =
kE (i) ξk2 ∀ξ ∈ H ⊗ h.
(0.5)
i∈I
1. Column spaces
A closed subspace V of B(H1 ; H2 ), for Hilbert spaces H1 and H2 , is called an
operator space in B(H1 ; H2 ). For each n ≥ 1, Mn (B(H1 ; H2 )) is naturally identified
(n)
(n)
with B(H1 ; H2 ), where H(n) denotes the n-fold orthogonal sum of copies of H.
In this way a norm is induced on Mn (V ). A bounded operator φ : V → W into an
operator space W in B(K1 ; K2 ) induces bounded operators
φ(n) : Mn (V ) → Mn (W );
[Tji ] 7→ [φ(Tji )];
φ is completely bounded (CB) if its complete bound kφkcb := supn≥1 kφ(n) k is finite.
Standard references for operator spaces and complete boundedness are Paulsen’s
book ([Pau]) and the review article by Christensen and Sinclair ([ChS]); two new
books are about to appear ([EfR],[Pis]). However, the proof of our main result is
elementary, depending only on the simple lemmas below.
For an operator space V in B(H1 ; H2 ) and a Hilbert space h, we define two
operator spaces, the column space Ch (V )b and the square-matrix space Mh (V )b :
Ch (V )b := {T ∈ B(H1 ; H2 ⊗ h) : E e T ∈ V ∀e ∈ h},
Mh (V )b := {T ∈ B(H1 ⊗ h; H2 ⊗ h) : E d T Ee ∈ V ∀d, e ∈ h}.
(1.1)
The column space Ch (V )b is an operator space in B(H1 ; H2 ⊗ h) lying between the
minimal tensor product V ⊗min h := V h (closure in the norm topology, [BlP]),
uw
and the ultraweak tensor product V ⊗uw h := V h . It equals the first if V or h is
finite dimensional and equals the second if and only if V is ultraweakly closed. The
analogous inclusions also hold for the square-matrix space. In particular, h = Cn
provides the identifications Mh (V )b = V ⊗min B(h) = V ⊗ Mn = Mn (V ).
Remark. Since the map e 7→ E e is linear and isometric, T ∈ B(H1 ; H2 ⊗ h) belongs
to Ch (V )b provided that E e T ∈ V for all vectors e from any total subset of h.
When A is a C ∗ -algebra the operator space Mh (A)b is typically not an algebra,
in particular the inclusion A ⊗min B(h) ⊂ Mh (A)b is strict. As an example to
illustrate this point take A = c, the commutative unital C ∗ -algebra of convergent
complex sequences —
represented on l2 by diagonal matrices, and let h = l2 . Let
L∞
2
T ∈ B(l ⊗ h) = B( n=1 l2 ) be given by the matrix


e1 0 0 · · ·
e3 0 0 · · ·


e5 0 0 · · · ,


.. .. .. . .
.
. . .
where ek = diag[0, . . . , 0, 1, 0, . . .] with 1
that T ∗ T has matrix

a 0
0 0

.. ..
. .
in the kth place and zeros elsewhere, so

···
· · ·
,
..
.
where a = diag[1, 0, 1, 0, . . .]. Then T ∈ Mh (A)b since ek ∈ c for each k, but
T ∗T ∈
/ Mh (A)b since a ∈
/ c. In particular T ∈ Mh (A)b \A ⊗min B(h).
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J MARTIN LINDSAY AND STEPHEN J WILLS
Lemma 1.1. Let V be an operator space in B(H1 ; H2 ) and let h and k be Hilbert
spaces. Then Ch (Ck (V )b )b = Ck⊗h (V )b .
Proof. By the associativity of the Hilbert space tensor product, both are subspaces
of B(H1 ; H2 ⊗ k ⊗ h). For S ∈ B(H1 ; H2 ⊗ k ⊗ h)
E e S ∈ Ck (V )b ∀e ∈ h ⇐⇒ E d,e S ∈ V ∀e ∈ h, d ∈ k, where E d,e = E d E e = E d⊗e
⇐⇒ E f S ∈ V ∀f ∈ k ⊗ h,
by the above remark. The result follows.
By the Parseval identity (0.5), a choice of orthonormal basis (ei )i∈I for h gives
L rise
to an identification of elements T of Ch (V )b with bounded operators H → i∈I H
defined by column matrices [E (i) T ]i∈I .
Lemma 1.2. Let V and W be operator spaces in B(H1 ; H2 ) and B(K1 ; K2 ) respectively, and let φ : V → W be a completely bounded map. Then for each Hilbert
space h there is a unique map φ(h) : Ch (V )b → Ch (W )b satisfying
E e φ(h) (T ) = φ(E e T )
∀e ∈ h, T ∈ Ch (V )b .
(1.2)
Moreover φ(h) is linear and bounded, and kφ(h) k ≤ kφkcb .
Proof. Pick an orthonormal basis (ei )i∈I of h and for each T ∈ Ch (V )b set T i =
E (i) T ∈ V , so that T is identified with the column matrix [T i ]i∈I . Let h00 be the
following dense subspace of h: {e ∈ h : hei , ei 6= 0 for only finitely many i}. For
each finite subset F = {i1 , . . . , in } of I, define the n × n operator matrix

 i
T 1 0 ··· 0

..
..  ,
T F :=  ...
.
.
T in
0 ···
0
(n)
thus T F ∈ Mn (V ) and kT F k ≤ kT k. For w ∈ K1 , letting w ∈ K1
column vector associated to (w, 0, . . . , 0), we have
X
kφ(T i )wk2 = kφ(n) (T F )wk2 ≤ (kφkcb kT kkwk)2 .
denote the
(1.3)
i∈F
L
Hence (φ(T i )w)i∈I is a well-defined element of i∈I K2 = K2 ⊗ h. Let φ(h) (T ) be
the linear map K1 → K2 ⊗ h defined by w 7→ (φ(T i )w)i∈I . Then for all e ∈ h00 ,
X
hu ⊗ e, φ(h) (T )wi =
hhei , eiu, φ(T i )wi = hu, φ(E e T )wi
i∈I
which shows that φ(h) is unique, and that (1.2) holds for such e. By continuity of
the maps e 7→ E e and φ, (1.2) holds for all e ∈ h, and (1.3) implies that φ(h) is
bounded and satisfies kφ(h) k ≤ kφkcb .
Remarks. φ(h) clearly extends the algebraic tensor product φ id : V h → W h,
and thus also the canonical extension of this map to the minimal tensor products of these spaces. In view of the natural (completely isometric) identification
of Mn (Ch (V )b ) and Ch (Mn (V ))b , k(φ(h) )(n) k = k(φ(n) )(h) k and so in fact φ(h) is
completely bounded and kφ(h) kcb = kφkcb .
Thus any completely bounded map between two operator spaces lifts to a completely bounded map between their column spaces. We now apply this construction
to completely bounded maps V → Ch (V )b .
EXISTENCE OF FELLER COCYCLES
5
Lemma 1.3. Let V be an operator space and h a Hilbert space with orthonormal
basis (ei )i∈I . Let (ψn )n≥1 be a sequence of completely bounded maps V → Ch (V )b ,
and for each i ∈ I and n ≥ 1 let ψni : V → V be the completely bounded map defined
by
ψni (T ) = E (i) ψn (T ).
Then, for each n ≥ 1,
ψnin ◦ · · · ◦ ψ1i1 = E (in ,...,i1 ) Ψn ◦ · · · ◦ Ψ1 ,
where E
(in ,...,i1 )
u
= E for u = ein ⊗ · · · ⊗ ei1 , and Ψk =
(H)
ψk
(1.4)
(k−1)
for H = ⊗
h.
Proof. For S ∈ C⊗(n−1) h (V )b ,
E (in ,...,i1 ) Ψn (S) = E (in ) E (in−1 ,...,i1 ) Ψn (S)
= E (in ) ψn (E (in−1 ,...,i1 ) S) = ψnin (E (in−1 ,...,i1 ) S).
Iterating this identity gives (1.4).
2. Existence of Feller Cocycles
Let A be a unital C ∗ -algebra acting on h and let η = (ei )i≥1 be an orthonormal
basis for a separable noise dimension space k, which is extended to a basis ηb =
(eα )α≥0 of b
k = C ⊕ k by putting e0 = 1. After introducing the relevant classes of
mapping matrix θ, quantum stochastic process on A, and Markovian cocycle on A,
we use the properties of column spaces established in the previous section to prove
the existence of completely positive contractive solutions of (0.1) by showing that
necessary conditions on θ for a solution to be completely positive and contractive
are sufficient for Picard iteration to yield a solution. Basis independence is then
confirmed, and this allows us to frame existence of solutions and characterisation
of cocycles as a bijective correspondence between a class of square-matrix space
valued maps on A (i.e. coordinate-free mapping matrices) and the space of regular
Markovian CP contraction cocycles on A, for a given noise dimension space.
Mapping matrices. The extended basis ηb of b
k is used to freely identify the Hilbert
L
spaces h ⊗ b
k and α≥0 h as in (0.5). A matrix θ = [θβα ] consisting of bounded linear
maps on A is called a mapping matrix. Such a matrix is regular if (0.2) holds, and
strongly regular if furthermore the suprema over n ≥ 1 and kak = kuk = 1 are finite
for each d; θ is (completely) bounded if there is a (completely) bounded operator
θe : A → B(h ⊗ b
k) such that
α
e
E (α) θ(a)E
(β) = θβ (a).
(2.1)
e
In the latter case, by the argument in the remark before Lemma 1.1, θ(A)
⊂ Mbk (A)b
and so θe may be viewed as an element of B(A; Mbk (A)b ). Finally θ has completely
bounded columns if for each β ≥ 0 there is a completely bounded map θβ : A →
B(h; h ⊗ b
k) satisfying
θβ (a)u = (θβα (a)u)α≥0 .
In this case effectively θβ ∈ CB(A; Cbk (A)b ). When θ is bounded
e
θeed : a 7→ E d θ(a)E
e,
d, e ∈ b
k
recovers the sesquilinear map Θ : (d, e) 7→ θeed mentioned in the introduction. The
collections of regular, strongly regular, bounded and completely bounded mapping matrices are denoted respectively MD (B(A))R , MD (B(A))SR , MD (B(A))b
and MD (B(A))cb . Thus B(A; Mbk (A)b ) and CB(A; Mbk (A)b ) are the coordinatefree counterparts to the last two classes of mapping matrix.
6
J MARTIN LINDSAY AND STEPHEN J WILLS
Processes on A ([Par],[LW1]). For a Borel subset I of R+ let FI denote the
symmetric Fock space over L2 (I; k), and write F for FR+ . The factorisation F =
FI ⊗ FI c is best seen through exponential vectors: ε(f ) = ε(f |I ) ⊗ ε(f |I c ); FI is
also identified with the following subspace of F: {ξ ⊗ ε(0) : ξ ∈ FI }. The definition
of processes requires the notions of admissible sets and adaptedness. For a subset
S of L2 (R+ ; k), ES denotes Lin{ε(f ) : f ∈ S}; S is admissible if ES is dense in F
and f[0,t] ∈ S whenever f ∈ S and t ≥ 0. A useful admissible set is
i
M[η] = {f ∈ (L2 ∩ L∞
loc )(R+ ; k) : f 6= 0 for only finitely many i},
where f i (·) := E (i) f (·) ∈ L2 (R+ ). An ES -process defined on A is then a family of
linear maps kt : A → L(h ES ; h ⊗ F) satisfying the adaptedness and measurability
conditions
(i) kt (a)uε(f[0,t] ) ∈ h ⊗ F[0,t] ; kt (a)uε(f ) = [kt (a)uε(f[0,t] )] ⊗ ε(f[t,∞[ );
(ii) t 7→ kt (a)uε(f ) is weakly measurable;
for all u ∈ h, f ∈ S, where uε(f ) abbreviates u ⊗ ε(f ). A natural further condition
to impose is
(iii) kt (a) is affiliated to A00 ⊗ B(F).
A bounded process is a process k for which kt (A) ⊂ B(h ES ; h ⊗ F) and a 7→ kt (a)
is bounded for each t. When the C ∗ -algebra is not a von Neumann algebra and k is
bounded we are interested in processes satisfying the stronger Feller-type condition:
(iii)0 kt (A) ⊂ MF (A)b .
The linear space of ES -processes on A is denoted P(A, ES ); processes being identified
when they weakly agree almost everywhere. The subspace of processes satisfying
the strong regularity condition
sup{kkt (a)uε(f )k : kak = kuk = 1, t ∈ [0, T ]} < ∞
for all f ∈ S and T > 0 is denoted PSR (A, ES ). For later use we now collect an
amalgam of results from [LW1].
Proposition 2.1. Let θ ∈ MD (B(A))R . Then Picard iteration yields a process
k θ ∈ P(A, EM[η] ) strongly satisfying (0.1); the map θ 7→ k θ is injective, and if
θ ∈ MD (B(A))SR then k θ ∈ PSR (A, EM[η] ). Moreover if θ is also bounded then
huε(f ), ktθ (a)vε(g)i is given by
Z t2
XZ t
hε(f ), ε(g)i
dtn · · ·
dt1 hu, φ1 ◦ · · · ◦ φn (a)vi
(2.2)
n≥0
0
0
for all u, v ∈ h and f, g ∈ M[η], where φi = θeed for d = (1, f (ti )), e = (1, g(ti )) ∈ b
k.
Existence and basis independence. Let (A, η, θ) be the data for a QSDE (0.1),
with A a unital C ∗ -algebra and each θβα bounded.
Theorem 2.2. If the mapping matrix θ has completely bounded columns then θ is
strongly regular.
Proof. Let θβ : A → Cbk (A)b be the completely bounded maps associated with θ
(β ≥ 0). By the three lemmas
θβαnn ◦ · · · ◦ θβα11 (a) = E (α) θβ (a)
(H)
where E (α) = E e for e = eα := eαn ⊗ · · · ⊗ eα1 and θβ = Ψn ◦ · · · ◦ Ψ1 for Ψk = θγ
where H = ⊗(k−1)b
k and γ = βk . Since {eα : α1 , . . . , αn ≥ 0} is an orthonormal
EXISTENCE OF FELLER COCYCLES
7
basis for ⊗(n)b
k, the Parseval identity (0.5) implies that
X
kθβαnn ◦ · · · ◦ θβα11 (a)uk2 = kθβ (a)uk2
α
≤ (kθβn kcb · · · kθβ1 kcb kakkuk)2 .
It follows that the constants (0.3) satisfy
Cnθ (a, u, d)2 ≤ (d + 1) max kθβ k2cb
n
0≤β≤d
kak2 kuk2 ,
thus θ is strongly regular.
Let φ ∈ CB(A; Mbk (A)b ). The following admissible set is clearly basis independent:
M(k) := {f ∈ (L2 ∩ L∞
loc )(R+ ; k) : dim Lin Rf < ∞},
where Rf is the essential range of f . Suppose that η1 and η2 are orthonormal
bases of k, then there is a third basis η3 such that M[η1 ] ∪ M[η2 ] ⊂ M[η3 ]; let θi
(i = 1, 2, 3) denote the mapping matrix obtained from φ and ηi such that φ = θei
according to (2.1). Then ktθ3 (a) extends both ktθ1 (a) and ktθ2 (a), by (2.2). It follows
that there is a process k φ ∈ PSR (A, EM(k) ) satisfying ktφ (a) ⊃ ktθ (a) whenever θ
is the coefficient matrix of φ with respect to some basis η. This gives rise to an
injective map
κ : CB(A; Mbk (A)b ) → PSR (A, EM(k) ),
φ 7→ k φ .
(2.3)
Remark. When k is finite dimensional, κ is naturally defined as a map B(A; A ⊗
B(b
k)) → PSR (A, E), where E = ES for S = (L2 ∩ L∞
loc )(R+ ; k) ([Eva]).
Feller cocycles. For a process k ∈ P(A, EM(k) ), each pair f, g ∈ M(k), and t ≥ 0,
define a map ktf,g : A → L(h) by
ktf,g (a) = E(f[0,t] )∗ kt (a)E(g[0,t] )
where E(h) := Eε(h) . The process k is a Markovian cocycle if each ktf,g is a bounded
map A → B(h), and the family satisfies
ktf,g (A) ⊂ A;
k0f,g = idA ;
s∗ f,s∗
rg
f,g
kr+t
= krf,g ◦ kt r
(2.4)
2
where (sr )r≥0 is the one-parameter semigroup of isometric right shifts on L (R+ ; k).
When A is a von Neumann algebra and k is CB and normal, (2.4) is equivalent
to (0.4) ([LW2], Section 4). In fact, by the use of square-matrix spaces, when k is
CB there is a natural definition of b
kt for which (2.4) is equivalent to (0.4) together
with the condition
kt (A) ⊂ MF[0,t] (A)b ,
k0 (a) = a ⊗ 1,
for any C ∗ -algebra A, or indeed any operator space ([LW3]).
When each of the families (ktf,g )t≥0 is strongly continuous in t, k is called a
Feller cocycle. This is relevant when A is a C ∗ -algebra rather than a von Neumann
algebra. A regular Markovian cocycle is one whose Markov semigroup (kt0,0 )t≥0 is
norm continuous in B(A). Regular Markovian contraction cocycles on a C ∗ -algebra
are automatically Feller ([LW2], Proposition 5.4).
Define the following class of basis independent mapping matrices:
ΦCPc (A, k) := {φ ∈ CB(A; Mbk (A)b ) : φ(a) + ∆(a) = ψ(a) + J ∗ aE (0) + E(0) aJ
and φ(1) ≤ 0, for some CP map ψ and bounded operator J},
where ∆(a) = a ⊗ P and P is the orthogonal projection of b
k onto k.
8
J MARTIN LINDSAY AND STEPHEN J WILLS
Theorem 2.3. For a unital C ∗ -algebra A acting on h, and a separable noise dimension space k, the map κ defined in (2.3) restricts to a bijection from ΦCPc (A, k) to
the space of regular Markovian CP contraction cocycles on A with noise dimension
space k.
Proof. Let φ ∈ CB(A; Mbk (A)b ) and let k = k φ . Then k is a regular Markovian cocycle by [LW2], Proposition 5.2. If φ lies in ΦCPc (A, k) then k is CP and contractive
by Proposition 5.1 and Theorem 5.2 of [LW1]. Thus κ restricts to an injection from
ΦCPc into the space of regular Markovian CP contraction cocycles. Surjectivity
follows from [LW2], Theorem 5.10 and [LW1], Proposition 5.1.
Remark. This result is a stochastic generalisation of a theorem of Christensen and
Evans ([ChE]).
References
[Bel]
[BlP]
[Bra]
[ChE]
[ChS]
[Con]
[EfR]
[EvH]
[Eva]
[Kun]
[LiP]
[LW1]
[LW2]
[LW3]
[Mey]
[MoS]
[Par]
[Pau]
[Pis]
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School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD
E-mail address: [email protected]
E-mail address: [email protected]