Dilation of a class of quantum dynamical semigroups with unbounded generators on UHF algebras

www.elsevier.com/locate/anihpb

Dilation of a class of quantum dynamical semigroups with unbounded generators on UHF algebras

Debashish Goswami, Lingaraj Sahu

, Kalyan B. Sinha

Stat-Math Unit, Indian Statistical Institute, 203, B.T. Road, Kolkata 700 108, India Received 19 November 2003; received in revised form 8 December 2004; accepted 14 December 2004

Dedicated to the memory of Professor Paul-André Meyer

Abstract

Evans–Hudson flows are constructed for a class of quantum dynamical semigroups with unbounded generator on UHF algebras, which appeared in [Rev. Math. Phys. 5 (3) (1993) 587–600]. It is shown that these flows are unital and covariant.

Ergodicity of the flows for the semigroups associated with partial states is also discussed.

2005 Elsevier SAS. All rights reserved.

Résumé

Les flots d’Evans–Hudson sont construits pour une classe de semi-groupes dynamiques quantiques à générateur non borné sur une algèbre UHF, définie dans la référence [Rev. Math. Phys. 5 (3) (1993) 587–600]. On montre que ces flots préservent l’unité et sont covariants. L’ergodicité des flots associés à des états partiels est également discutée.

2005 Elsevier SAS. All rights reserved.

1. Introduction

Quantum dynamical semigroups, to be abbreviated as QDS, constitute a natural generalization of classical Markov semigroups arising as expectation semigroups of Markov processes. A QDS{Tt:t0}on aC^∗-algebraA is aC₀-semigroup of completely positive mapsT_t onA. Given such a QDS, it is interesting and important to look for a dilation in the sense of Evans–Hudson, i.e. a family of∗-homomorphismsη_t:A→A⊗B(Γ (L²(R₊,k₀)))

* Corresponding author.

E-mail addresses: goswamid@isical.ac.in (D. Goswami), lingaraj_r@isical.ac.in (L. Sahu), kbs@isical.ac.in (K.B. Sinha).

1 The author would like to acknowledge the support of National Board of Higher Mathematics, DAE, India and to the DST-DAAD programme.

2 The author would like to acknowledge the support from Indo-French Centre for the Promotion of Advanced Research as well as from DST-DAAD programme.

0246-0203/$ – see front matter 2005 Elsevier SAS. All rights reserved.

doi:10.1016/j.anihpb.2004.12.001

where k₀is some separable Hilbert space andΓ (·)denotes the symmetric Fock space, satisfying a suitable quan- tum stochastic differential equation. This problem has been completely solved for QDS with bounded generators by Goswami, Sinha and Pal [2,4], where a canonical Evans–Hudson flow for an arbitrary QDS with bounded gen- erator has been constructed. However, only partial success has been achieved for QDS with unbounded generator.

It is perhaps too much to expect a complete general theory for an arbitrary QDS. It may be wiser to look for Evans–

Hudson flow for special classes of QDS. In [3] for example, the authors gave a general theory of dilation for QDS on aC^∗-algebraA, which is covariant with respect to an action of a Lie group and also symmetric with respect to a given faithful semifinite trace. However, in the present article, we shall try to construct an Evans–Hudson flow for another class of QDS on a UHFC^∗-algebra, studied by T. Matsui in [6]. This construction has some similarity with the earlier one, but the action of the discrete groupZ^dinstead of a Lie group action as in [3] makes the present model somewhat different from that of [3]. We have not only proved the existence of a dilation in Section 3, we are also able to prove in Section 4 that the Evans–Hudson (EH) flow is indeed covariant with respect to theZ^daction.

Some ergodicity properties of the flows are also discussed briefly in Section 5.

2. Notation and preliminaries

T. Matsui [6] constructed a class of conservative QDS on the UHFC^∗-algebraAgenerated as theC^∗-completion of infinite tensor product

j∈Z^d MN(C), where N andd are two fixed positive integers. ThisC^∗-algebra can also be described as the inductive limit of full matrix algebras{M_Nⁿ(C), n1}with respect to the imbedding M_Nⁿ⊆M_Nn+1 by sendinga toa⊗1. The unique normalized trace tr on Ais given by tr(x)=_N¹nTr(x), for x∈M_Nⁿ(C), where Tr denotes the ordinary trace onM_Nⁿ(C). For x∈M_N(C)andj ∈Z^d, letx^{(j )} denote an element inAwhosejth component isx and rest are identity ofM_N(C). For a simple tensor elementa∈A, let a(j )be thejth component ofa. The support ofa, denoted by supp(a)is defined to be the set{j ∈Z^d: a(j )=1}. For a general elementa∈Asuch thata=_∞

n=1c_na_nwitha_n’s simple tensor elements inAandc_n’s complex coefficients, we define supp(a):=

n∈Nsupp(a_n)and we set|a| =cardinality of supp(a). For anyΛ⊆Z^d, let AΛ denote the∗-subalgebra generated by elements ofAwith supportΛ. WhenΛ= {k}, we writeAk instead of A{k}. Let Aloc be the ∗-subalgebra of Agenerated by elements a∈A of finite support or equivalently by {x^{(j )}: x∈M_N(C), j ∈Z^d}. ClearlyAlocis dense inA. Fork∈Z^d, the translationτ_konAis an automorphism determined byτ_k(x^{(j )}):=x^(j+k)∀x∈M_N(C)andj∈Z^d. Thus, we get an actionτ of the infinite discrete group Z^d onA. For x∈Awe denoteτ_k(x)byx_k. The algebraAis naturally sitting inside h₀=L²(A,tr), the GNS Hilbert space for(A,tr). It is easy to see thatτ_k extends to a unitary on h0, to be denoted by the same symbolτ_k, giving rise to a unitary representationτ of the groupZ^don h₀, which implements the actionτ. It is also clear that this action extends as an action ofZ^dby normal automorphisms on the von Neumann algebraA.

We also need another dense subset ofA, which is in a sense like the first Sobolev space inA. For this, we need to note thatM_N(C)is spanned by a pair of noncommutative representatives{U, V}of ZN= {0,1, . . . , N−1} such thatU^N =V^N=1∈MN(C)and U V =wV U, wherew∈Cis the primitive Nth root of unity. These U, V can be chosen to be theN ×N circulant matrices. In particular forN =2, a possible choice is given by U=σ_x andV =σ_z, whereσ_x andσ_z denote the Pauli-spin matrices. Forj ∈Z^d and(α, β)∈G≡ZN×ZN, we setσ_j;α,β(x)= [U^{(j )α}V^{(j )β}, x] ∀x∈A,x1=

j;α,βσ_j;α,β(x) andC¹(A)= {x∈A: x1<∞}. It is easy to see thatx^∗1= τ_j(x)1= x1 and since C¹(A)contains the dense ∗-subalgebra Aloc, C¹(A)is a denseτ invariant ∗-subalgebra ofA. Let G:=

j∈Z^dGbe the infinite direct product of the finite group Gat each lattice site. Thus each g∈G has jth componentg_{(j )}=(α_j, β_j)withα_j, β_j ∈ZN. For g∈G we define its support by supp(g)= {j ∈Z^d: g_{(j )}=(0,0)}and|g| =cardinality of supp(g). Let us consider the projective unitary representation ofG, given by Gg→U_g=

j∈Z^dU^{(j )αj}V^{(j )βj} ∈A. For a given completely positive mapT onA, we formally define the Linbladian

L=

k∈Z^d

Lk,

whereLkx=τ_kL0(τ_−kx), ∀x∈A, withL0(x)= −1

2

T (1), x

+T (x), (2.1)

and{A, B} :=AB+BA.

In particular we consider the LindbladianLfor the completely positive mapT, T x:=

∞ n=0

a^∗_nxa_n, ∀x∈A,

associated with a sequence of elements{an}n0inA, withan=

g∈Gcn,gUgsuch that_∞

n=0

g∈G|cn,g||g|²<

∞. Matsui has proven the following in the paper referred earlier [6].

Theorem 2.1. (i) The mapLformally define above is well defined onC¹(A)and the closureLˆofL/_C1(A) is the generator of a conservative QDS{P_t: t0}onA,

(ii) The semigroup{P_t}leavesC¹(A)invariant.

The semigroupP_t satisfies Pt(x)=x+

t

0

Ps Lˆ(x)

ds, ∀x∈Dom(Lˆ).

Since 1∈C¹(A)andLˆ(1)=L(1)=0, it follows thatP_t(1)=1,∀t0.

Following [6], we say thatP_t is ergodic if there exists an invariant stateψsatisfying

P_t(x)−ψ (x)1→0 ast→ ∞, ∀x∈A. (2.2)

In [6], the author has discussed some criteria for ergodicity of the QDSP_t. Some examples of such semigroups associated with partial states on the UHF algebra and their perturbation are given.

For a state φ on M_N(C) andk∈Z^d, the partial state φ_k on A is determined by φ_k(x)=φ(x_(k))x_{k}c, for x=x_(k)x_{k}c, where x_(k)∈Ak and x_{k}c ∈A{k}^c. We can find a natural number N and elements {L^(m): m= 1,2, . . . , N}inMN(C)such that

φ(x)=

N

m=1

L^(m)∗xL^(m) ∀x∈M_N(C) and

N

m=1

L^(m)∗L^(m)=1.

Form=1, . . . , N, let us consider the elementL^(m)₀ ∈A0with the zeroth component beingL^(m). Now fork∈Z^d andm=1, . . . , N, writingL^(m)_k =τk(L^(m)₀ ), the partial stateφk is given by,

φ_k(x)=

N

m=1

L^(m)_k ^∗xL^(m)_k ∀x∈A.

By (2.1), the LinbladianL^φcorresponding to the partial stateφ₀is formally given by L^φ(x)=

k∈Z^d

L^φ_k(x),

where

L^φ_k(x)=φ_k(x)−x=1 2

N

m=1

L^(m)_k ^∗, x

L^(m)_k +L^(m)_k ^∗ x, L^(m)_k

.

It follows from Theorem 2.1 thatL^φ is defined on C¹(A). Moreover, the closure Lˆ^φ of L^φ/C¹(A)generates a conservative QDSP_t^φonAgiven by

P_t^φ

k∈Λ

x_(k)

=

k∈Λ

φ(x_(k))+e^−t x_(k)−φ(x_(k)) .

We note that the mapΦ defined by, Φ

k∈Λ

x_(k)

= lim

t→∞P_t^φ

k∈Λ

x_(k)

=

k∈Λ

φ(x_(k))

extends as a state onAwhich is the unique invariant state for the ergodic QDSP_t^φ. For any real numberc, we consider the perturbation

L^(c)(x)=L^φ(x)+cL(x), ∀x∈C¹(A).

It is clear thatL^(c)is the Linbladian associated with the completely positive map T (x)=

N

m=1

L^(m)_k ^∗xL^(m)_k +c ∞

l=0

a_l^∗xal, ∀x∈A

and by Theorem 2.1 it follows that the closureLˆ^(c)ofL^(c)/C¹(A)generate a QDSP_t^(c). Moreover, one has Theorem 2.2 [6]. There exists a constantc₀ such that for 0cc₀ the above QDSP_t^(c) is ergodic with the invariant stateΦ^(c)satisfying

P_t^(c)(x)

12e^{−(1−c/c0)t}x1,

(2.3) P_t^(c)(x)−Φ^(c)(x)1 4

N²e^{−(1−c/c0)t}x1, ∀x∈C¹(A).

Remark 2.3. The invariant stateΦ^(c)corresponding to the ergodic QDSP_t^(c)is given by Φ^(c)(x)=Φ(x)+c

∞

0

Φ L P_t^(c)(x)

dt, ∀x∈C¹(A).

Let us conclude the present section with a brief discussion on the fundamental integrator processes of quantum stochastic calculus, introduced by Hudson and Parthasarathy [5]. Let k=L²(R₊,k₀)where k₀=l²(Z^d)with the canonical orthonormal basis{ej: j ∈Z^d}andΓ =Γ_sym(k), the symmetric Fock space over k. Forf ∈k, we denote by e(f )the exponential vector inΓ associated withf:

e(f )=

n0

√1 n!f⁽ⁿ⁾, wheref⁽ⁿ⁾=f⊗f ⊗ · · · ⊗f

n-copies

forn >0 and by conventionf⁽⁰⁾=1. Forf =0, e(f )is called the vacuum vector inΓ. LetC be the space of all bounded continuous functions fromR+ to k₀, so thatE(C)≡ {e(f ): f ∈C}is

total inΓ (k). Anyf ∈L²(R₊,k₀)decomposes asf =

k∈Z^df_ke_k withf_k∈L²(R₊). We take the freedom to use the same symbolf_k to denote the function inL²(R₊,k₀)as well, whenever it is clear from the context. The fundamental processes{Λ^j_i: i, j∈Z^d}associated with the orthonormal basis{e_j: j∈Z^d}are given by

Λⁱ_j(t )=a_χ[0,t]⊗ei fori=0, j =0

=a_χ^†_[0,t]⊗ej fori=0, j=0

=Λ_Mχ

[0,t]⊗|e_je_i| fori, j=0

=t1 fori=j =0,

whereM_χ[0,t]is the multiplication operator onL²(R₊)by characteristic function of the interval[0, t]. For details the reader is referred to [10] and [7].

3. Evans–Hudson type dilation

In this section we investigate the possibility of constructing EH flows for the QDS on UHFC^∗-algebra, discussed in the previous section. Although the question is not answered in full generality, EH flows for a class of QDS are constructed.

Letr=

g∈Gc_gU_g∈Asuch that

g∈G|c_g||g|²<∞. The LindbladianLassociated with the elementr, i.e.

associated with the CP mapT , T (x)=r^∗xr,∀x∈A, takes the form L(x)=

k∈Z^d

δ_k^†(x)r_k+r_k^∗δ_k(x), (3.1)

wherer_k:=τ_k(r)andδ_k, δ_k^†are bounded derivation onAgiven by δ_k(x)= [x, r_k] and δ_k^†(x):= δ_k(x^∗)_∗

= [r_k^∗, x], ∀x∈A. (3.2)

It follows from [6] that the closure Lˆ of L/C¹(A) is the generator of a contractive QDS Pt on A. In or- der to construct an EH flow for the QDS Pt, we would like to solve the following QSDE in B(L²(A,tr))⊗ B(Γ (L²(R₊,k₀))):

dj_t(x)=

j∈Z^d

j_t δ_j^†(x)

da_j(t )+

j∈Z^d

j_t δ_j(x)

da_j^†(t )+j_t Lˆ(x) dt,

(3.3) j₀(x)=x⊗1_Γ, x∈Aloc.

Let us first look at the corresponding Hudson–Parthasarathy equation inL²(A,tr)⊗Γ (L²(R+,k0)), given by dU_t=

j∈Z^d

r_j^∗da_j(t )−r_jda^†_j(t )

−1 2

j∈Z^d

r_j^∗r_jdt

U_t,

(3.4) U₀(x)=1_L2⊗Γ.

However, though eachr_j∈Aand hence is inB(L²(A,tr)), Eq. (3.4) does not in general admit a solution since

u,

j∈Z^d

r_j^∗rju

=

j∈Z^d

rju² ∀u∈L²(A,tr),

is not convergent in general and hence

j∈Z^dr_j⊗e_j does not define an element inA⊗k₀. For example, letrbe the single-supported unitary elementU^(k)∈Afor somek∈Z^d so thatr_j =U^{(k+j )} is a unitary for eachj ∈Z^d and hence

j∈Z^d

rju²=

j∈Z^d

u²= ∞.

However, as we shall see, in many situation there exist Evans–Hudson flows, even though the corresponding Hudson–Parthasarathy equation (3.4) does not admit a solution.

Remark 3.1. There are some cases when an Evans–Hudson flow can be seen to be implemented by a solution of a Hudson–Parthasarathy equation. For example, given a self adjointr∈A

dVt=Vt k∈Z^d

S_k^∗dak(t )−Skda_k^†(t )

−1 2

k∈Z^d

S_k^∗Skdt

, V0=1,

whereS_kis defined byS_k(x)= [r_k, x]forx∈A⊆L²(A,tr), admits a unique unitary solution and x→V_t(x⊗1)V_t^∗

gives an Evans–Hudson dilation forP_t [8,9].

Leta, b∈ZN be fixed and W =U^aV^b∈MN(C). We consider the following representation of the infinite product groupG:=

j∈Z^dZN, given by Gg→W_g=

j∈Z^d

W^{(j )αj}, whereg=(α_j).

For anyy∈A,y=

g∈Gc_gU_gand forn1 we define ϑ_n(y)=

g∈G

|c_g||g|ⁿ. Now we considerr∈A, r=

g∈Gc_gW_g such that

g∈G|c_g||g|²<∞. It is clear thatϑ₁(r)=

g∈G|c_g||g|<

∞. We note that anyx∈Aloccan be written asx=

h∈Gc_hU_h, with complex coefficientsc_h satisfyingc_h=0 for allhsuch that supp(h)∩supp(x)is empty. So

ϑ_n(x)=

h∈G

|c_h||h|ⁿ<∞ forn1, and it is clear that

ϑ_n(x)|x|ⁿ

h∈G

|c_h|cⁿ_x wherec_x= |x|(1+

h∈G|c_h|). Let us consider the formal LindbladianLassociated with the elementr, L=

k∈Z^d

Lk,

whereLk(x)=¹₂δ_k^†(x)r_k+r_k^∗δ_k(x).

Forn1, we denote the set of integers{1,2, . . . , n}byI_n and for 1pn, P = {l₁, l₂, . . . , l_p} ⊆I_n with l₁< l₂<· · ·< l_p, we define a map from then-fold Cartesian product ofZ^dto that ofpcopies ofZ^dby

k(I¯ n)=(k₁, k₂, . . . , kn)→ ¯k(P ):=(kl₁, kl₂, . . . , kl_p)

and similarly,ε(P )¯ :=(ε_l1, ε_l2, , . . . , ε_lp)for a vectorε(I¯ _n)=(ε₁, ε₂, , . . . , ε_n)in then-fold Cartesian product of {−1,0,1}.

For brevity of notations, we writeε(P )¯ ≡c∈ {−1,0,1}to mean that allε_li =cand denote k(I¯ _n)andε(I¯ _n) byk(n)¯ andε(n)¯ respectively. Settingδ^ε_k=δ_k^†,Lkandδ_kdepending uponε= −1,0 and 1 respectively, we write R(¯k)=r_k1r_k2· · ·r_kp andδ(k,¯ ε)¯ =δ^ε_k^p

p· · ·δ_k^ε1

1 for anyk¯=(k₁, k₂, . . . , k_p)andε¯=(ε₁, ε₂, . . . , ε_p). Now we have the following useful lemma,

Lemma 3.2. Letr, xand constantcxbe as above. Then (i) For anyn1,

¯ k(n)

δ k(n),¯ ε(n)¯

(x) 2ϑ₁(r)cx

n

∀x∈Aloc, whereε(n)¯ is such thatεl=0, ∀l∈In.

(ii) For anyn1 andk(n),¯ Lk_n· · ·Lk₁(x)= 1

2ⁿ

p=0,1,...,n

P⊆In:|P|=p

R k(P¯ ^c)_∗

δ k(n),¯ ε¯(P )(n)

(x)R k(P )¯ ,

whereε¯_{(P )}(n)is such thatε¯_{(P )}(P )≡ −1 andε¯_{(P )}(P^c)≡1.

(iii) For anyn1, pn, P⊆I_nandε(n)¯ such thatε(P )¯ contains all those components equal to 0, we have,

¯ k(n)

δ k(n),¯ ε(n)¯ (x)r^p 2ϑ₁(r)c_xn

1+ rn

2ϑ₁(r)c_xn

.

(iv) Letm₁, m₂1;x, y∈Alocandε¯(m₁),ε¯(m₂)be two fixed tuples. Then forn1 andε(n)¯ as in (iii), we have,

k(n),¯ k¯(m1),¯k(m2)

δ k(n),¯ ε(n)¯

δ k¯(m₁),ε¯(m₁)

(x)·δ k¯(m₂),ε¯(m₂)(y) 2ⁿ 1+ r2n+m₁+m₂

2ϑ₁(r)c_x,yn+m₁+m₂

, wherec_x,y=max{c_x, c_y}.

Proof. (i) Asr^∗is again of the same form asr, it is enough to observe the following:

k_n,...,k₁

[r_kn,· · · [r_k1, x]] · · ·] 2ϑ₁(r)c_xn

∀x∈Aloc.

In order to prove this let us consider

LHS=

kn,...,k₁

g_n,...,g₁∈G;h∈G

|c_gn| · · · |c_g1||c_h|[τ_knW_gn,· · · [τ_k1W_g1, U_h]] · · ·].

We note that for any two commuting elementsA, B inA,[A, [B, x]] = [B, [A, x]]. Thus, for the commutator [τ_knW_gn,· · · [τ_k1W_g1, U_h]] · · ·]to be nonzero, it is necessary to have(supp(g_i)+k_i)∩supp(h)=φfor eachi= 1,2, . . . , n. Clearly the number of choices of suchk_i∈Z^dis at most|g_i| · |h|. Thus we get,

kn,...,k1

[rk_n,· · · [rk₁, x]] · · ·]

g_n,...,g₁∈G;h∈G

|cg_n| · · · |cg₁||ch||gn| · · · |g1||h|ⁿ2ⁿ 2ϑ1(r)cx

n

.

(ii) The proof is by induction. For anyk∈Z^dwe have, Lk(x)=1

2

k∈Z^d

δ^†_k(x)r_k+r_k^∗δ_k(x),

so it is trivially true for n=1. Let us assume it to be true for some m >1 and for any k_m+1∈Z^d consider Lkm+1Lkm· · ·Lk1(x). By applying the statement forn=mwe get,

Lkm+1Lkm· · ·Lk1(x)= 1 2^m+1

p=0,1,...,m

P⊆Im:|P|=p

δ_k^∗

m+1

R k(P¯ ^c)_∗

δ k(m),¯ ε¯_{(P )}(m)

(x)R k(P )¯ r_km+1 +r_k^∗

m+1δ_km+1

R k(P¯ ^c)_∗

δ k(m),¯ ε¯_{(P )}(m)

(x)R k(P )¯ . Sincer_k’s are commuting with each other, the above expression becomes

1 2^m+1

p=0,1,...,m

P⊆I_m:|P|=p

R k(P¯ ^c)_∗ δ_k^∗

m+1δ k(m),¯ ε¯_{(P )}(m)

(x)R k(P )¯ r_km+1 +r_k^∗

m+1R k(P¯ ^c)_∗

δ_km+1δ k(m),¯ ε¯_{(P )}(m)

(x)R k(P )¯

= 1 2^m+1

p=0,1,...,m+1

P⊆Im+1:|P|=p

R k(P¯ ^c)_∗

δ k(m¯ +1),ε¯_{(P )}(m+1)

(x)R k(P )¯ .

(iii) By simple application of (ii), δ k(n),¯ ε(n)¯

(x)= 1 2^p

q=0,1,...,p

Q⊆P:|Q|=q

R k(P¯ \Q)_∗

δ k(n),¯ ε¯_{(Q,P )}(n)

(x)R k(Q)¯

, (3.5)

whereε¯_{(Q,P )}(n)is defined to be the map from then-fold Cartesian product of{−1,0,1}to itself, given byε(n)¯ →

¯

ε_{(Q,P )}(n)such thatε¯_{(Q,P )}(Q)≡ −1,ε¯_{(Q,P )}(P\Q)≡1 andε¯_{(Q,P )}(In\P )= ¯ε(In\P ). Now (iii) follows from (i).

(iv) By (3.5) we have, LHS= 1

2^p

¯

k(n),k¯(m₁),¯k(m₂)

q=0,1,...,p

Q⊆P:|Q|=q

R k(P¯ \Q)_∗

×δ k(n),¯ ε¯(Q,P )(n)

δ k¯(m₁),ε¯(m₁)

(x)·δ k¯(m₂),ε¯(m₂) (y)

R k(Q).¯ Now applying the Leibnitz rule, it can be seen to be less than or equal to

r^p 2^p

k(n),¯ k¯(m1),¯k(m2)

q=0,1,...,p

Q⊆P:|Q|=q

l=0,1,...,n

L⊆I_n:|L|=l

δ k(L),¯ ε¯_{(Q,P )}(L)

δ k¯(m₁),ε¯(m₁)(x)

×δ k(L¯ ^c),ε¯_{(Q,P )}(L^c)

δ k¯(m₂),ε¯(m₂)(y).

Using (iii), we obtain, LHS(1+ r)ⁿ

2^p

q=0,1,...,p

p! (p−q)!q!

l=0,1,...,n

n!

(n−l)!l! 1+ rl+m1 2ϑ₁(r)c_xl+m1

× 1+ rn−l+m₂

2ϑ₁(r)c_yn−l+m₂

2ⁿ 1+ r2n+m₁+m₂

2ϑ₁(r)c_x,yn+m₁+m₂

. 2

Now we are in a position to prove the following result about existence of an Evans–Hudson flow for QDSPt

associated with the elementr∈Adiscussed above.

Theorem 3.3. (a) Fort0, there exists a unique solutionj_tof the QSDE, djt(x)=

j∈Z^d

jt(δ_j^†x)daj(t )+

j∈Z^d

jt(δjx)da_j^†(t )+jt(Lˆx)dt, (3.6) j₀(x)=x⊗1_Γ, ∀x∈Aloc,

such thatjt(1)=1, ∀t0.

(b) Forx, y∈Alocandu, v∈h0, f, g∈C, ue(f ), j_t(xy)ve(g)

=

j_t(x^∗)ue(f ), j_t(y)ve(g)

. (3.7)

(c)j_textends uniquely to a unitalC^∗-homomorphism fromAintoA⊗B(Γ ).

Proof. We note first thatAlocis a dense∗-subalgebra ofA.

(a) As usual, we solve the QSDE by iteration. Fort₀0, tt₀andx∈Aloc, we set j_t⁽⁰⁾(x)=x⊗1_Γ and

(3.8) j_t⁽ⁿ⁾(x)=x⊗1_Γ +

t

0

j∈Z^d

j_s⁽ⁿ⁻¹⁾ δ^†_j(x)

da_j(s)+

j∈Z^d

j_s⁽ⁿ⁻¹⁾ δ_j(x)

da_j^†(s)+j_s⁽ⁿ⁻¹⁾ Lˆ(x) ds.

Then foru∈h₀andf ∈C, we can show by induction, that j_t⁽ⁿ⁾(x)−j_t⁽ⁿ⁻¹⁾(x)ue(f )(t₀cf)^n/2

√n! ue(f )

k(n)¯

¯ ε(n)

δ k(n),¯ ε(n)¯ (x), (3.9)

wherec_f =2e^γf(t0)(1+ f²_∞), withγ_f(t₀)=_t0

0 (1+ f (s)²)ds. Forn=1, by the basic estimate of quantum stochastic integral [10,7],

j_t⁽¹⁾(x)−j_t⁽⁰⁾(x)

ue(f )²

=

t 0

j∈Z^d

δ^†_j(x)da_j(s)+

j∈Z^d

δ_j(x)da_j^†(s)+ ˆL(x)ds

ue(f )

2

2e^γf(t0)e(f )^2t

0 j∈Z^d

δ^†_j(x)u²+

j∈Z^d

δ_j(x)u²+Lˆ(x)u² 1+f (s)²ds

c_ft₀e(f )²

j∈Z^d

δ_j^†(x)u+δ_j(x)u+Lj(x)u2

.

Thus (3.9) is true forn=1. Inductively assuming the estimate for somem >1, we have by the same argument as above,

j_t^(m+1)(x)−j_t^(m)(x)

ue(f )²

=

t 0

j∈Z^d

j_s^(m)

m δ^†_j(x)

−j_s^(m−1)

m δ_j^†(x)

da_j(s_m)+

j∈Z^d

j_s^(m)

m δ_j(x)

−j_s^(m−1)

m δ_j(x)

da_j^†(s_m)

+ j_s^(m)

m Lˆ(x)

−j_s^(m−1)

m Lˆ(x) ds_m

ue(f )

2

2e^γf(t0) t

0 j∈Z^d

j_s^(m)

m δ_j^†(x)

−j_s^(m−1)

m δ_j^†(x)ue(f )²

+

j∈Z^d

j_s^(m)

m δ_j(x)

−j_s^(m−1)

m δ_j(x)ue(f )² +j_s^(m)

m Lˆ(x)

−j_s^(m−1)

m Lˆ(x)ue(f )² 1+f (s_m)² ds_m c_f

t

0 j∈Z^d

j_s^(m)

m δ_j^†(x)

−j_s^(m−1)

m δ_j^†(x) ue(f )

+

j∈Z^d

j_s^(m)

m δ_j(x)

−j_s^(m−1)

m δ_j(x)

ue(f )+j_s^(m)

m Lˆ(x)

−j_s^(m−1)

m Lˆ(x)

ue(f ) ²ds_m. Now applying (3.9) forn=m, we get the required estimate for n=m+1 and furthermore by the estimate of Lemma 3.2(iii),

j_t⁽ⁿ⁾(x)−j_t⁽ⁿ⁻¹⁾(x)

ue(f )3ⁿ(t₀c_f)^n/2

√n! ue(f ) 1+ rn

1+2ϑ₁(r)cx

n

.

Thus it follows that the sequence{j_t⁽ⁿ⁾(x)ue(f )}is Cauchy. We definej_t(x)ue(f )to be lim_n→∞j_t⁽ⁿ⁾ue(f ), that is

j_t(x)ue(f )=xu⊗e(f )+

n1

j_t⁽ⁿ⁾(x)−j_t⁽ⁿ⁻¹⁾(x)

ue(f ) (3.10)

and one has

j_t(x)ue(f )ue(f )

x +

n1

3ⁿ(t₀c_f)^n/2

√n! 1+ rn

1+2ϑ₁(r)c_xn

. (3.11)

Uniqueness follows by setting, q_t(x)=j_t(x)−j_t(x) and observing

dq_t(x)=

j∈Z^d

q_t δ^†_j(x)

da_j(t )+

j∈Z^d

q_t δ_j(x)

da_j^†(t )+q_t L(x)

dt, q₀(x)=0.

Exactly similar estimate as above shows that, for alln1, q_t(x)ue(f )(t₀c_f)^n/2

√n! ue(f )

¯ k(n)

¯ ε(n)

δ k(n),¯ ε(n)¯ (x).

Since by Lemma 3.2(iii) the sum grows asnth power,q_t(x)=0∀x∈Aloc, showing the uniqueness of the solution.

As 1∈AlocwithLk(1)=δ_k^†(1)=δ_k(1)=0 it follows from the QSDE (3.6) thatj_t(1)=1.

(b) Forue(f ), ve(g)∈h⊗E(C)andx, y∈Aloc, we have, by induction, j_t⁽ⁿ⁾(x^∗)ue(f ), ve(g)

=

ue(f ), j_t⁽ⁿ⁾(x)ve(g) . Now asntends to∞, we get

jt(x^∗)ue(f ), ve(g)

=

ue(f ), jt(x)ve(g) .