Journal of Applied Analysis Vol. 4, No. 2 (1998), pp. 143–160
FLOWS OF QUANTUM NOISE ROBIN L. HUDSON, J. MARTIN LINDSAY and K.R. PARTHASARATHY Received June 5, 1997 and, in revised form, January 5, 1998
Abstract. The dynamics of classical and quantum systems in the presence of noise is usually described in the language of stochastic differential equations. When the system observables comprise a C ∗ -algebra, stochastic evolutions are obtained by solving such equations driven by creation, preservation and annihilation processes on Fock space, with linear maps on the algebra as coefficients. ∗ -Homomorphic evolutions are obtained precisely when the collection of maps has a certain structure; this structure admits a cohomological description. Here we consider equations governing the joint evolution of the system and noise (from input to output) by supposing that the characteristics of the (input) noise processes are given by a group representation. The structure required for ∗ -homomorphic evolution is determined.
0. Introduction Quantum stochastic differential equations were originally used to obtain dilations of quantum dynamical semigroups on the full algebra of bounded operators on a Hilbert space ([9], [10]). This was done in two stages: first the Lindblad form ([13]) of the generator of such a semi–group was used to write 1991 Mathematics Subject Classification. 81S25, 81P15, 60H20. Key words and phrases. Quantum stochastic, quantum filtering, quantum measurement, input noise, multiple quantum Wiener integral.
ISSN 1425-6908
c Heldermann Verlag.
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down a QSDE whose solution provides a stochastic dilation; this was then combined with the time shift of the quantum noise to obtain an automorphic group dilation ([8], [15]). The group law is coded as a cocycle relation for the stochastic dilation, and it has been shown that (under various regularity conditions) all such adapted cocycles arise as solutions of QSDE’s of this form ([11], [1], [4]). These stochastic dilations have the form jt (a) = u∗t aut ;
dut = l βα ut dΛ βα (t);
u0 = 1
where [Λ βα ] is the matrix of QS integrators: creation–preservation–annihilation–time ([17]), and [l βα ] is a matrix of bounded Hilbert space operators for which there are necessary and sufficient algebraic conditions to guarantee unitarity of (ut ). Moreover it follows from the quantum Itˆo formula that the process (jt ) itself satisfies a QSDE: djt = jt ◦ θ βα dΛ βα (t);
j0 (a) = a ⊗ 1
(0.1)
where ∗
∗
θ βα (a) = l αβ a + al βα + ljα aljβ (summation j = 1, . . . d). The unitarity of (ut ) implies that (jt ) is ∗ -homomorphic and unital. Viewing (0.1) as an equation for a stochastic flow on a general unital C ∗ -algebra, necessary and sufficient conditions on the matrix of bounded linear maps [θ βα ] were found for existence and uniqueness of a ∗ -homomorphic unital solution ([6]): θ βα (a∗ b) = θ αβ (a)∗ b + a∗ θ βα (b) + θjα (a)∗ θjβ (b);
θ βα (1) = 0
(again summation j = 1, . . . , d). In scattering theory and quantum filtering, interest focuses on the evolution of the field, or the quantum noise ([18], [3], [19], [2]). The unitary stochastic dilations can readily be applied by conjugation. The question therefore arises: is there a more intrinsic description of this in the spirit of (0.1)? Specifically can one (a) formulate QSDE’s for the evolution of a quantum noise (“output process”), (b) characterise the coefficients which lead to homomorphic evolutions, and (c) determine conditions under which the evolution is compatible with an evolution of the algebra? Here we give one such formulation, in which the evolving noise has a description in terms of a representation of an underlying group, and solve both the above characterisation problems under a boundedness assumption on the QSDE coefficients. Our results generalise [6]. Our tools for the analytic part are the multiple quantum Wiener integral and basic quantum stochastic calculus ([16], [17]). The algebraic part comprises an elementary cohomological analysis of structure relations for the QSDE coefficients. The main result is Theorem 3.4 in
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conjunction with Theorem 2.1 (existence and uniqueness) and Theorem 4.7 (the solution of the structure relations). 1. Notations and preliminaries Let (Λ αβ ) be the basic integrators of quantum stochastic calculus ([17]). Thus (Λ01 , . . . , Λ0d ), (Λ10 , . . . , Λd0 ) and [Λij ]1≤i,j≤d are respectively the multidimensional creation, annihilation and preservation processes, and Λ00 denotes time. We shall adopt the Einstein convention in which repeated indices are summed over (unless otherwise indicated), Greek indices α, β, γ, δ will run from 0 to d and Roman indices i, j, k will run from 1 to d, and we supplement these with a further index which runs through α : α, β = 0, 1, . . . , d . (1.1) Ξ := η = β † Ξ carries the involution αβ = αβ . These latter indices will be used in the following way: for η = αβ , L η denotes the QS integrand L αβ and Λ η
denotes the QS integrator Λ βα . Thus Z d Z X η L η dΛ = L αβ dΛ βα
†
Λ η = (Λ η )∗ .
and
α,β=0
Let H denote the tensor product of a fixed separable Hilbert space h and symmetric Fock space over L2 (R+ ; R), where R is a finite dimensional Hilbert space governing the dimension of the quantum noise. In this paper the domain of all QS integrals will be the algebraic tensor product h ⊗ E where E is the linear span of the exponential vectors εϕ : ϕ ∈ L2 ∩ L∞ loc (R+ ; R)}. Note that h ⊗ E is dense in H. An adapted QS process L will be Rt called QS-integrable if L(·)x is Borel measurable and 0 kL(s)xk2 ds < ∞ for each x ∈ h ⊗ E. Let {F η , G η : η ∈ Ξ} be adapted QS-integrable processes. The fundamental relations of the calculus are ([9]): Rt Rt for X(t) = X(0) + 0 F η (s) dΛ η (s) and Y (t) = Y (0) + 0 G η (s) dΛ η (s), Z t huε ϕ , [X(t) − X(0)]vε ψ i = huε ϕ , F η (s)vε ψ i χ η (s) ds 0 (QSC1) k[X(t) − X(0)]vε ϕ k2 ≤ K(t, ϕ) max η
Z
hX(t)uε ϕ , Y (t)vε ψ i − hX(0)uε ϕ , Y (0)vε ψ i =
t
0
Z
0
kF η (s)vε ϕ k2 ds t
(QSC2)
X(s)uε ϕ , Gαβ (s)vε ψ
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D E
o + F αβ (s)uε ϕ , Y (s)vε ψ + F αi (s)uε ϕ , Giβ (s)vε ψ χ βα (s) ds,
(QSC3)
where χ η (s) = χ βα (s) = ϕ α (s)ψ β (s) = ϕ α (s)ψ β (s), and K(t, ϕ) = et k(d) maxα kϕ α k2L∞ ([0,t]) . We shall need some multiple QS integrals. Let ∆n be the simplex {s ∈ (R+ )n : s1 > · · · > sn } and let ∆n[0,t] = {s ∈ ∆n : t > s1 }. Definition. Let D be a dense subspace of h. A map F from ∆n into L(D; h) (the space of linear maps from D into h) is strongly locally integrable if F (·)v|∆n[0,t] is (Bochner) integrable for each v ∈ D and t > 0. Its multiple quantum Wiener integrals are then defined through their matrix elements: with domain D ⊗ E, * + Z Z uε ϕ , F (s)dΛη (s) vε ψ = hu, F (s)vi χ η (s) ds hε ϕ , ε ψ i , ∆n [0,t]
where χ η (s) =
∆n [0,t]
Qn
i=1 χ
ηi (s ). i
By Pettis’ Theorem, and the density of D ⊗ E in H, Borel measurability for a map f : R+ → H is equivalent to measurability of each of the maps {hx, f (·)i : x ∈ D ⊗ E}. Using this the following result is easily established. Lemma 1.1. If F : ∆n → L(D; h) is strongly locally square integrable then the iterated QS integrals Z t Z sn−1 η1 X(t) := dΛ (s1 ) · · · dΛ ηn (sn )F (s), η ∈ Ξn , 0
0
are well-defined, coincide with the corresponding multiple quantum Wiener integral, and moreover satisfy the estimate Z 2 kϕk2 n kX(t)vε ϕ k ≤ e K(t, ϕ) kF (s)vk2 ds. n ∆[0,t] (QSC20 )
2. Existence and uniqueness In this section we describe the class of quantum stochastic differential equation (QSDE) systems to be considered and give a general existence and uniqueness theorem. Let G be a topological group, A a unital C ∗ -algebra acting on the Hilbert space h and Ξ the index set (1.1). An involution on G × Ξ × A is given by (g, η, a)† = (g −1 , η † , a∗ ). Let G be a subgroup of the group of Borel measurable maps from R+ into G (under pointwise multiplication), which
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is closed under multiplication by indicator functions of intervals, so that pt Φ := χ[0,t] Φ ∈ G whenever Φ ∈ G. Denote the identity element of G by E. We consider maps θ : G × Ξ × A → A satisfying the following conditions with respect to G: Φ(s )
Φ(s )
(Ai) Each of the maps s ∈ ∆n 7→ θ ηn n ◦ · · · ◦ θ η1 1 (a)v ∈ h is Borel measurable, where θgη (a) denotes θ[g, η, a]. (Aii) For each Φ ∈ G there is a locally bounded function α Φ on R+ satisfying
Φ(s)
θ η (a) ≤ α Φ (s) kakA . A
Without loss we assume that α Φ is non-decreasing and bounded below by 1.
Associated with such a map θ, is the following system of QSDE’s: η djtΦ (a) = jtΦ ◦ θΦ(t) η (a)dΛ (t);
j0Φ (a) = a,
(2.1)
in which summation over repeated indices is understood and jtΦ (a) denotes jt (Φ, a). A solution of the system (2.1) is a map j from R+ × G × A into L, the linear space of operators on H with domain h ⊗ E, such that • (jt (Φ, θ [Φ(t), η, a]))t≥0 is an adapted QS-integrable process, Rt • jt (Φ, a) = a + 0 js (Φ, θ[Φ(s), η, a]) dΛ η (s) for each t, Φ and a.
With one dimension of quantum noise these are some of the basic examples that we have in mind. (a) G trivial, (i) θ11 (a) = wa − a; θ01 (a) = la; θ10 (a) = −l∗ wa; θ00 (a) = (ih − 12 l∗ l)a, where w, l, h ∈ A, h is self-adjoint and w is unitary. In this case jt (a) = ut a where (ut )t≥0 is a unitary-valued QS process of the kind considered in [9]. ∗ (ii) θ11 = σ−id.; θ01 a σ-derivation; θ10 = (θ01 )† : a 7→ θ01 (a∗ ) ; θ00 a linear involution–preserving map satisfying θ00 (ab) − θ00 (a)b − aθ00 (b) = θ01 (a)θ10 (b) where σ is a *-homomorphism on A. Then (jt )t≥0 is a QS flow on the algebra of the kind considered in [7], [6]. (b) A trivial, G = (C × T × R, ◦) with group law (l1 , w1 , x1 ) ◦ (l2 , w2 , x2 ) = (l1 + w1 l2 , w1 w2 , x1 + x2 − Im(¯l1 w1 l2 ))
and G = L2loc (R+ ; C) × M (R+ ; T) × L1loc (R+ ; R), where M here denotes the (measure equivalence classes of) measurable functions. (i) For g = (z, eiθ , x), θ11 (g, a) = (eiθ − 1)a; θ01 (g, a) = za; 2 |z| 0 θ10 (g, a) = −¯ z eiθ a and θ0 (g, a) = ix − a. 2
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In this case jt (Φ, a) = WtΦ a where, if Φ = (f, u, h) then WtΦ = ei
Rt 0
h(s) ds
W (f χ[0,t] )Γ(Muχ[0,t] ).
Here W is the Fock representation of the canonical commutation relations over L2 (R+ ) ([5], [17]), Γ is the second quantisation map and M denotes multiplication operator on L2 (R+ ). −1 −1 (ii) jt (Φ, a) = WtΨ WtΦ WtΨ a = WtΨΦΨ a, Ψ a fixed element of G. In this case θ βα (g, a) may be computed from (i) and the group identity (w, eiψ , t)(z, eiθ , s)(w, eiψ , t)−1 = (eiψ z + (1 − eiθ )w, eiθ , s + Im[|w|2 eiθ − wze ¯ iψ (1 + e−iθ )]).
(c) A a unital C ∗ -algebra, G and G as in (b),
jt (Φ, a) = ut WtΦ au∗t
where (ut ) is the unique solution of the QSDE dut = ut (ldA∗ + (v − 1)dΛ − l∗ vdA + (ih − 12 l∗ l)dt);
u0 = 1
in which l, v, h ∈ A with v unitary and h self-adjoint. In this case, for g = (z, eiθ , s), θ11 (g, a) = (eiθ vav ∗ − a)
θ01 (g, a) = (1 + zv)a − eiθ vav ∗ l
θ10 (g, a) = (al∗ − eiθ z¯a − eiθ l∗ vav ∗ )
θ00 (g, a) = i[h, a] − 21 {l∗ l, a} + l∗ v(1 − eiθ )a + eiθ z¯av ∗ l + eiθ l∗ vav ∗ l |z|2 a. + is − 2
We seek to generalise example (c) in the spirit in which (a)(ii) generalises the restriction of (c) to A: θ11 (a) = vav ∗ − a; θ10 (a) = la − vav ∗ l; 1 ∗ ∗ ∗ θ0 (a) = al − l vav ; θ00 (a) = i[h, a] − 12 {l∗ l, a} + l∗ vav ∗ l
(cf. [12]).
Theorem 2.1. Let θ : G × Ξ × A → A be a map satisfying the conditions (Ai) and (Aii). (i) The system of QSDE’s (2.1) has a solution satisfying where
kjt (Φ, a)vε ϕ k ≤ γ Φ,ϕ (t)||a||||vε ϕ ||, γ Φ,ϕ (t) = α Φ (0) +
X
n≥1
1
(n!)− 2
np
on tK(t, ϕ)α Φ (t) .
(2.2)
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(ii) The system (2.1) has at most one solution (jt )t≥0 such that each jtΦ := jt (Φ, ·) is contractive, for each Φ. Proof. Let θ satisfy (Ai) and (Aii). Suppose that j (1) and j (2) are two
(i)
solutions of (2.1) satisfying the contractivity assumption jt (Φ, a) ≤ B(H)
||a||A for each application of the estimate (QSC2) to the Φ ∈ G. Repeated (1) (2) difference kt := jt − jt gives
kkt (Φ, a)vε ϕ k2 Z 1
2
1) ≤ K(t, ϕ) ds1 max ks1 Φ, θ ηΦ(s (a) vε ϕ 1 η 1 0 Z t Z sn−1
2
Φ(s1 ) n n) ◦ · · · ◦ θ (a) vε ≤ K(t, ϕ) ds1 · · · dsn max ksn Φ, θ ηΦ(s ϕ η1 n ≤
2 n!
(
0
0
p
tK(t, ϕ) sup α Φ (s) [0,t]
η
)2n
||a||2 ||vε ϕ ||2 .
Letting n → ∞ therefore establishes the uniqueness part. To establish the existence of a solution define a sequence of maps k (n) : R+ × G × A → L as follows: k (0) (t, Φ, a) = a; Z (n) k (t, Φ, a) =
∆n [0,t]
n) 0) θ ηΦ(s ◦ · · · ◦ θ ηΦ(s (a)dΛ(s, η) n 1
n = 1, 2, . . . .
The assumptions (Ai) and (Aii), together with Lemma 1.1, imply that the maps k (n) are well-defined and satisfy (i0 ) (ii0 )
(n)
t 7→ kt (Φ, θ[Φ(t),η, a]) is an adapted QS-process; R t (n) (n+1) k (Φ, a) = 0 ks (Φ, θ[Φ(s), η, a])dΛ η (s);
t
(n)
(iii0 ) kt (Φ, a)vε ϕ ≤ γnΦ,ϕ (t)||a||||vε ϕ ||; on np tK(t, ϕ) sup[0,t] α Φ (s) for n ≥ 1 and γ0Φ,ϕ (t) = where γnΦ,ϕ (t) = (n!)−1/2 α Φ (0). Therefore
jt (Φ, a)vε ϕ =
X
(n)
kt (Φ, a)vε ϕ
n≥0
defines a family of QS-processes (jt (Φ, a))t≥0 such that (jt (Φ,θ[Φ(t), η, a]))t≥0 is QS-integrable. Moreover, by (ii0 ), the identity (QSC1) and an application
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of Lebesgue’s Dominated Convergence Theorem, Z t η js (Φ, θ[Φ(s), η, a]) dΛ vε ψ uε ϕ , 0 Z t = huε ϕ , js (Φ, θ[Φ(s), η, a])vε ψ i χ η (s) ds 0 E XZ tD = uε ϕ , ks(n) (Φ, θ[Φ(s), η, a])vε ψ χ η (s) ds n≥0 0
=
E XD (n+1) uε ϕ , kt (Φ, a)vε ψ
n≥0
= huε ϕ , [jt (Φ, a) − a] vε ψ i ,
where we used the notation χ η (s) = ϕα (s)ψβ (s) for η = αβ . Therefore the QSDE system (2.1) is satisfied, and the estimate (2.2) follows from (iii0 ). We shall refer to the solution constructed above as the Picard solution, and (2.2) as the Picard estimate. Notice that we have not yet used the group structure — the existence and uniqueness theorem requires merely that G is a set of maps, from R+ into some parameter set, which is invariant under multiplication by indicator functions of the intervals [0, t]. 3. Structure relations In this section we demonstrate the way in which algebraic properties of the maps (jt ) are inherited from corresponding properties of θ. The goal is to establish necessary and sufficient conditions on θ for (jt ) to be linear and unital on A, and involutive and multiplicative on G × A. We first find necessary conditions under the following assumption on G: (Aiii ) ∀g ∈ G∃Φ ∈ G, T > 0 such that Φ(s) = g for s < T. Proposition 3.1. Let G satisfy (Aiii) and suppose that j is a solution of the system of QSDE’s (2.1).
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(i) (ii) (iii) (iv)
If If If If
each each each each
151
jtΦ is linear then each θgη is linear. jtE is unital then θeη (1) = 0 ∀g, η. jt is involutive then θ is involutive. jt is adjoint–multiplicative: jt (Φ−1 , a∗ )∗ jt (Ψ, b) = jt (ΦΨ, ab),
then θ satisfies θ βα [gh, ab] = (θ αβ [g −1 , a∗ ])∗ b + aθ βα [h, b] + (θiα [g −1 , a∗ ])∗ θiβ [h, b].
(3.1)
(iv0 ) If each jt is contractive and multiplicative, then θ satisfies θ βα [gh, ab] = θ βα [g, a]b + aθ βα [h, b] + θiα [g, a]θiβ [h, b].
(3.2)
In (iv) and (iv0 ) summation over i = 1, . . . , d is understood.
Proof. Let g, h ∈ G, a, b ∈ A, λ ∈ Φ(s) = g and Ψ(s) = h for s < T .
C and fix Φ, Ψ ∈ G, T
> 0 such that
(i) If each jtΦ is linear then, for t < T , Z t jsΦ (θgη (a + λb) − θgη (a) − λθgη (b)) dΛ η (s) = 0
[jtΦ (a + λb) − jtΦ (a) − λjtΦ (b)] = 0.
(ii) If each jtE is unital then, for t < T , Z t jsE ◦ θeη (1) dΛ η (s) = jtE (1) − 1 = 0. 0
(iii) If each jt is involutive then, for t < T , Z tn o −1 −1 jsΦ (θ[g −1 , η, a∗ ]) − jsΦ (θ[g, η † , a]∗ ) dΛ(s, η) = 0
jt (Φ−1 , a∗ ) − jt (Φ, a)† = 0.
(iv) If j is adjoint-multiplicative then, by (QSC3), if t < T , Z t uε ϕ , jsΦΨ (θ βα [gh, ab]) dΛ βα (s)vε ψ D n0 o E = uε ϕ , jtΦψ (ab) − ab vε ψ D −1 E = jtΦ (a∗ )uε ϕ , jtΨ (b)vε ψ
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Z t nD E −1 = jsΦ (θ αβ [g −1 , a∗ ])uε ϕ , jsΨ (b)vε ψ 0 D −1 E + jsΦ (a∗ )uε ϕ , jsΨ (θ βα [h, b])vεψ Eo D −1 Φ i −1 ∗ Ψ i + js (θ α [g , a ])uε ϕ , js (θ β [h, b])vε ψ ϕ α ψ β (s) ds Z tD n uε ϕ , jsΦψ ((θ αβ [g −1 , a∗ ])∗ b) + jsΦΨ (aθ βα [h, b]) = 0 +jsΦΨ ((θiα [g −1 , a∗ ])∗ θiβ [h, b]) vε ψ ϕ α ψ β (s) ds.
Thus, applying (QSC1) once more, Z tn jsΦΨ (θ βα [gh, ab]) − jsΦΨ ((θ αβ [g −1 , a∗ ])∗ b) − jsΦΨ (aθ βα [h, b]) 0 −jsΦΨ ((θiα [g −1 , a∗ ])∗ θiβ [h, b]) dΛ βα (s) = 0
for t < T . (iv0 ) If each jtΦ is contractive then the Quantum Itˆo Lemma may be applied directly, and multiplicativity of each jt implies the identity Z t ΦΨ α js (θ β [gh, ab]) − jsΦΨ (θ βα [g, a]b) − jsΦΨ (aθ βα [h, b]) 0 −jsΦΨ (θiα [g, a]θiβ [h, b]) dΛ βα (s) = 0
for t < T . In each case independence of the quantum stochastic integrators ([14]) and h ⊗ E-weak continuity of the integrands implies that the integrand vanishes identically on [0, T ). Evaluating the integrands at s = 0 therefore yields the results since g, h, a, b and λ were arbitrary. Theorem 3.2. Let θ : G × Ξ × A → A and G satisfy (Ai) and (Aii), and let j be the Picard solution of the system of QSDE’s (2.1). Then (i) jtΦ is linear provided each θgη is linear. (ii) jtE is unital provided each θeη (1) = 0. (iii) jt is involutive provided θ is involutive. (iv) jt is adjoint-multiplicative provided each θgη is linear, θ is involutive, and θ satisfies (3.2). Proof. The linear, unital and involutive properties of the multiple quantum (n) Wiener integrals kt (n ≥ 1) are easily seen to be inherited from the corresponding properties of θ. (i), (ii) and (iii) easily follow. Under the assumptions of (iv) jt is involutive, so it suffices to prove adjoint-multiplicativity. β Fix Φ, Ψ ∈ G, u, v ∈ h and ϕ, ψ ∈ (L2 ∩ L∞ loc )(R+ ), let χ α = ϕ β ψ α and
Flows of quantum noise (1)
write mt = mt given by
(2)
− mt
(i)
where mt
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are the sesquilinear maps A × A →
C
(1) mt (a, b) = hjt (Φ, a)uε ϕ , jt (Ψ, b)vε ψ i ; m2t (a, b) = uε ϕ , jt (Φ−1 Ψ, a∗ b)vε ψ . On the one hand (QSC3) implies that the identity Z tn (l) α mt (a, b) − hauε ϕ , bvε ψ i = m(l) s (a, θ β [Ψ(s), b]) 0 o l β (l) i +ms (θ α [Φ(s), a], b) + ms (θ α [Φ(s), a], θiβ [Ψ(s), b]) χ βα (s) ds
(3.3)
holds for l = 1. On the other hand, by (QSC1) and the assumptions on θ, (3.3) is also satisfied for l = 2. Thus |mt (a, b)| is bounded by Z t Φ(s) c(t) max ms (θ ζ (a), θ ηΨ(s) (b)) ds, ζ,η∈Ξ•
0
where c is the locally bounded function t 7→ (d+1)2 (d+2) max sup |ϕ¯ β ψ α (s)|, α,β [0,t]
Ξ• = Ξ ∪ {•} and |mt (a, b)| ≤ n
c(t)
max
ζ,η∈(Ξ• )n
Z
θ•g (a)
∆n [0,t]
= a. Iterating this estimate gives
Φ(s ) Φ(s ) Φ(s1 ) n) ◦ · · · ◦ θ (b) msn θ ζn n ◦ · · · ◦ θ ζ1 1 (a), θΨ(s ds. ηn η1
By the Picard estimate (2.2), the integrand is bounded by n o n −1 γ Φ,ϕ (t)γ Ψ,ψ (t) + γ Φ Ψ,ψ (t) α Φ (t)α Ψ (t) ||a|| ||b|| ||uε ϕ || ||vε ψ || so, letting n → ∞, we obtain mt (a, b) = 0. Thus
hjt (Φ, a)uε ϕ , jt (Ψ, b)vεψ i = uε ϕ , jt (Φ−1 Ψ, a∗ b)vε ψ
foreach t, Φ, . . . , ψ. Since the domain of jt (Φ, a) is precisely the linear span d of uε ϕ : u ∈ h, ϕ ∈ (L2 ∩ L∞ loc )(R+ ; C ) , the result follows. Corollary 3.3. Under the linear, involutive and multiplicative assumptions on θ, together with (Ai) and (Aii), the Picard solution (jt ) of the system of QSDE’s (2.1) also has the contractive property ||jtΦ (a)|| ≤ ||a|| ∀ a ∈ A, Φ ∈ G, t ≥ 0. Proof. Since jt is adjoint multiplicative ||jt (Φ, a)x||2 = hx, jt (E, a∗ a)xi ≤ ||x|| ||jt (E, a∗ a)x||
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for each x ∈ h ⊗ E, where PNE is the identity element of G. Iterating this inequality gives, for x = i=1 vi ε ϕi , ||jt (Φ, a)x|| ≤ ||x||1/2 ||jt (E, a∗ a)x||1/2
n
n−1
n
≤ ||x||1/2 ||x||1/4 · · · ||x||1/2 ||jt (E, (a∗ a)2 )x||1/2 ( )2−n N X n−1 . ≤ ||x|| ||x||−1 ||vi ε ϕi ||γ E,ϕi ||(a∗ a)2 || i=1
Taking a limsup (n → ∞) and using the C ∗ -property of || · ||A therefore gives ||jt (Φ, a)x|| ≤ ||x||. Since h ⊗ E is dense in H, the result follows. Combining these results we have Theorem 3.4. Let A be a C ∗ -algebra and let G be a group of Borel maps from R+ into a topological group G satisfying the minimality condition (Aiii). Let θ : G × Ξ × A → A be a map satisfying the measurability condition (Ai) and the boundedness condition (Aii). Then the Picard solution (jt ) of the system of QSDE’s djt (Φ, a) = jt (Φ, θ[Φ(t), η, a]) dΛ η (t);
j0 (Φ, a) = a
is linear, unital, involutive and adjoint multiplicative if and only if θ is linear and unital in A, involutive on G × Ξ × A, and satisfies the identity θ βα [gh, ab] = θ βα [g, a]b + aθ βα [h, b] + θiα [g, a]θiβ [h, b],
in which case the solution is also contractive, and multiplicative.
4. Solution of the structural equations In the previous section we saw that if a flow on a system+noise algebra is governed by a QSDE system of the form djt (Φ, a) = jt (Φ, θ βα [Φ(t), a])dΛ βα (t), subject to assumptions (Ai–iii), then the maps (θ βα ) must satisfy the structure relations: θ βα (g, ·) is linear
(4.1)
θ βα (e, 1) = 0
(4.2)
θ βα (g, ·)† = θ αβ (g −1 , ·)
(4.3)
θ βα [gh, ab] = θ βα [g, a]b + aθ βα [h, b] + θiα [g, a]θiβ [h, b],
(4.4)
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and conversely, if (4.1–4.4) are satisfied then subject to assumptions (Ai–iii), the system n o has a unique solution which defines a flow. We shall call any solution θ βα of (4.1–4.4) a collection of structure maps. The structure equations are elucidated by introducing the following vector/matrix notation: θ α (g, a) = (θ1α (g, a), . . . , θdα (g, a)) ∈ Ad (row vector);
θ β (g, a) = [θ β (g, a)]> (column vector); σji (g, a) = θji (g, a) + aδji ; θ(g, a) = θji (g, a) ij ∈ Md (A); σ(g, a) = σji (g, a) i,j ∈ Md (A).
We also write θ βα (g) for θ βα (g, 1), θ βα (a) for θ βα (e, a), σ(g) for σ(g, 1), and so on. Proposition 4.1. The structure equations (4.4) imply: θ βα (ab) = θ βα (a)b + aθ βα (b) + θ α (a)θ β (b)
(4.4A)
θ βα (gh) = θ βα (g) + θ βα (h) + θ α (g)θ β (h)
(4.4G)
[θ βα (g), a] = θ α (a)θ β (g) − θ α (g)θ β (a), equivalently
(4.4A–G)
θ βα (g)a + θ α (g)θ β (a) = aθ βα (g) + θ α (a)θ β (g). Proof. (4.4A) and (4.4G) are particular cases of (4.4), and (4.4A–G) arises from two ways in which one can write θ βα (g, a) : θ βα (a) + aθ βα (g) + θ α (a) θ β (g) = θ βα (eg, aI) θ βα (ge, Ia) = θ βα (g) a + θ βα (a) + θ α (g) θ β (a)
Corollary 4.2. The structure maps satisfy: θ α (g) (a ⊗ 1) − aθ α (g) = θ α (a) θ (g) − θ α (g) θ (a) θ β (g) a − (a ⊗ 1) θ β (g) = θ (a) θ β (g) − θ (g) θ β (a) σ (a) σ (g) = σ (g) σ (a) . Proof. Putting β = i in (4.4A–G) gives the first identify, and putting α = j in (4.4A–G) gives the second. Putting β = i and α = j in (4.4A–G) gives θ (g) (a ⊗ 1) − (a ⊗ 1) θ (g) = θ (a) θ (g) − θ (g) θ (a) = (σ (a) − a ⊗ 1) θ (g) − θ (g) (σ (a) − a ⊗ 1) ,
so σ (a) θ (g) = θ (g) σ (a), and the third identity follows.
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Corollary 4.3. The structure maps satisfy σ (g, a) = σ (g) σ (a) where σ|G is a unitary group representation; σ|A is a *-algebra representation; and σ|G commutes with σ|A . Proof. Straightforward verification. Corollary 4.4. The structure maps satisfy θ0 (g, a) = θ0 (a) + σ (a) θ0 (g) where θ0 |G is a 1-cocycle with respect to σ : θ0 (gh) = θ0 (g) + σ (g) θ0 (h) ; θ0 |A is a σ-derivation: a linear map A → Ad satisfying θ0 (ab) = θ0 (a) b + σ (a) θ0 (b) ; and, for each g ∈ G, [I − σ (g)] θ0 is the inner σ-derivation a 7→ θ0 (g) a − σ (a) θ0 (g) . Proof. Straightforward verification. Corollary 4.5. The structure maps satisfy θ00 (g, a) = θ00 (a) + aθ00 (g) + θ0 (a) θ0 (g) where θ00 |A is linear, involutive and has coboundary (a, b) 7→ θ0 (a) θ0 (b) ;
θ00 |G is involutive and satisfies
θ00 (gh) − θ00 (g) − θ00 (h) = θ0 (g) θ0 (h) ;
and for each g ∈ G, θ0 (·) θ0 (g) − θ0 (g) θ0 (·) is the inner derivation a 7→ θ00 (g) , a . Proof. Straightforward verification. We may now prove the converse of Proposition 4.1.
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Proposition 4.6. Let A θ βα : A → A and (4.4A), (4.4G), (4.4A–G) and α A θ β (1)
α Gθ β
157
: G → G be maps satisfying
=G θ βα (e) = 0.
(4.2A/G)
Then θ βα (g, a) :=A θ βα (a) + a G θ βα (g) + (A θ α (a)) (G θ β (g)) defines maps θ βα : G × A → A satisfying (4.4) and θ βα |G×{1} =G θ βα ;
θ βα |{e}×A =A θ βα .
(*)
(4.5)
Proof. Under conditions (4.2A/G), (*) defines maps satisfying (4.5); it is therefore safe to drop the pre-subscripts. On the one hand, θ βα (ab) + abθ βα (gh) + θ α (ab) θ β (gh)
= θ βα (a) b + aθ βα (b) + θ α (a) θ β (b) + ab θ βα (g) + θ βα (h) + θ α (g) θ β (h) + III where III = {θiα (a) b + aθiα (b) + θ α (a) θi (b)} θiβ (g) + θiβ (h) + θi (g) θ β (h) = {θ α (a) σ (b) + aθ α (b)} {θ β (g) + σ (g) θ β (h)} .
On the other hand, θ βα (g, a) b + aθ βα (h, b) + θ α (g, a) θ β (h, b) = θ βα (a)+aθ βα (g)+θ α (a) θ β (g) b+a θ βα (b)+bθ βα (h)+θ α (b) θβ (h) + III 0
where III 0 = {θiα (a) + aθiα (g) + θ α (a) θi (g)} θiβ (b) + bθiβ (h) + θi (b) θ β (h) = {θ α (a) σ (g) + aθ α (g)} {θ β (b) + σ (b) θ β (h)}
Therefore, after cancelling θ βα (a) b + aθ βα (b) , we have θ βα (gh, ab) − θ βα (g, a) b − aθ βα (h, b) − θ α (g, a) θ β (h, b) = a b θ βα (g) + θ βα (h) + θ α (g) θ β (h) + θ α (b) {θ β (g) + σ (g) θ β (h)} +θ α (a) {θ β (b) + σ (b) [θ β (g) + σ (g) θ β (h)]} −a θ βα (g) b + θ βα (h) + θ α (b) θ β (h) + θ α (g) [θ β (b) + σ (b) θ β (h)] −θ α (a) {θ β (g) b + σ (g) [θ β (b) + σ (b) θ β (h)]} ,
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which may be expressed as a b, θ βα (g) + {θ α (b) θ β (g) − θ α (g) θ β (b)}
+ {bθ α (g) (b ⊗ 1) + θ α (b) θ (g) − θ α (g) θ (b)} θ β (h)} +θ α (a) {(b ⊗ 1) θ β (g) − θ β (g) b + θ (b) θ β (g) − θ (g) θ β (b)}
since σ (b) and σ (g) commute. But this vanishes, by Corollary 4.2, so the result follows. We may now draw these results together and display the full solution of the structure equations of a QS flow on system + noise algebra. The solution divides into three parts – the Evans-Hudson conditions on the algebra maps, complementary conditions on the group maps and consistency conditions between the two. Theorem 4.7. Let A σ : A → Md (A) be a unital ∗ -homomorphism, let d 0 d A θ0 : A → A (column vectors) be a A σ-derivation and let A θ0 : A → A 0 be a linear involution whose co-boundary is (a, b) 7→ A θ (a) (A θ0 (b)). Let d G σ : G → Md (A) be a unitary (group) representation, let G θ0 : G → A be a 0 1-cocycle with respect to σ, and let G θ0 : G → A be an involution satisfying θ00 (gh) − θ00 (g) − θ00 (h) = θ0 (g) θ0 (h). Then, putting θ (a) =Aσ (a) − a ⊗ 1, ∗ θ (g) =G σ (g) − 1, θ0 (a) = [A θ0 (a∗ )]∗ and θ0 (g) = G θ0 g −1 , θ βα (g, a) := θ βα (a) + aθ βα (g) + θ α (a) θ β (g)
defines a solution of the structure equations for a QS flow if and only if (1) A σ and G σ commute; (2) a 7→ [I − σ (g)]A θ0 (a) coincides with the inner σ-derivation a 7→ θ0 (g) a − σ (a) θ0 (g); (3) A θ0 (·) θ0 (g)A θ0 (·) coincides with the inner derivation a 7→ θ00 (g) , a .
Proof. Under the above conditions θ βα (e, 1) = 0 and (4.4) is satisfied, by Proposition 4.6. Moreover the linear and involutive conditions on θ βα are easily verified: θ00 g −1 , a∗ = θ00 (a∗ ) + a∗ θ00 g −1 + θ0 (a∗ ) θ0 g −1 ∗ = θ00 (a) + θ00 (g) a + θ0 (g) θ0 (a) ∗ = θ00 (a) + θ00 (g) a + θ0 (a) θ0 (g) − θ00 (g) a + aθ00 (g) ∗ = θ00 (g, a) .
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Acknowledgement. Part of this work was done during a Royal Society funded visit of KRP to Nottingham, and part during a visit of JML to New Delhi under the Exchange Agreement between the Indian National Science Academy and the Royal Society.
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Robin L. Hudson Mathematics Department University of Nottingham Nottingham NG7 2RD, UK email:
[email protected]
J. Martin Lindsay Mathematics Department University of Nottingham Nottingham NG7 2RD, UK email:
[email protected]
K.R. Parthasarathy Indian Statistical Institute (Delhi Centre) 7 SJS Sansanwal Marg New Delhi 110016, India email:
[email protected]
