White Noise on Bialgebras (Lecture Notes in Mathematics, 1544) 3540566279, 9783540566274

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Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, Zurich F. Takens, Groningen

1544

Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, Zurich F. Takens, Groningen

1544

Michael Schiirmann

White Noise

on Bialgebras

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest

Author Michael Schurmann Institut fur Angewandte Mathematik Universitat Heidelberg 1m Neuenheimer Feld 294 W-6900 Heidelberg, Germany

Mathematics Subject Classification (1991): 81S25, 81R50, 60130, 81S05, 60B15 ISBN 3-540-56627-9 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-56627-9 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1993 Printed in Germany Typesetting: Camera-ready by author/editor 46/3140-543210 - Printed on acid-free paper

To Jutta, Antje and Matthias

CONTENTS

Introduction 1. Basic concepts and first results 1.1. Preliminaries 1.2. Quantum probabilistic notions 1.3. Independence 1.4. Commutation factors 1.5. Invariance of states 1.6. Additive and multiplicative white noise 1.7. Involutive bialgebras 1.8. Examples 1.9. White noise on involutive bialgebras 2. Symmetric white noise on Bose Fock space 2.1. Bose Fock space over L2 (R.+, H) 2.2. Kernels and operators 2.3. The basic formula 2.4. Quantum stochastic integrals and quantum Ito's formula 2.5. Coalgebra stochastic integral equations 3. Symmetrization 3.1. Symmetrization of bialgebras 3.2. Schoenberg correspondence 3.3. Symmetrization of white noise 4. White noise on Bose Fock space 4.1. Group-like elements and realization of white noise 4.2. Primitive elements and additive white noise 4.3. Az ema noise and quantum Wiener and Poisson processes 4.4. Multiplicative and unitary white noise 4.5. Co commutative white noise and infinitely divisible representations of groups and Lie algebras 5. Quadratic components of conditionally positive linear funetionals 5.1. Maximal quadratic components 5.2. Infinitely divisible states on the Weyl algebra 6. Limit theorems 6.1. A coalgebra limit theorem 6.2. The underlying additive noise as a limit 6.3. Invariance principles REFERENCES SUBJECT INDEX

1 12 12 15 17 19 21 22 26 30 35 41

41 45 52 57 65 69 69 74 79 81 81 85 90

94 103 114 114 122 128 128 130 132 138 143

Introduction These notes are a contribution to the field of quantum (or non-commutative] probability theory. Quantum probability can be regarded as an attempt of a unified approach to classical probability and the quantum theory of irreversible processes. Our special interest lies in a non-commutative theory of processes with independent and stationary increments on a group. Following common practise, such processes will be called 'White noise even though they are actually 'integrated white noise'. The motivating idea is to use non-commutative white noise as a description of a quantum mechanical heat bath to which the quantum mechanical initial system under consideration is coupled. The passage from classical commutative to quantum non-commutative theory is mathematically established by replacing algebras of functions by algebras of linear operators, i.e. by not necessarily commutative algebras. The natural generalization of an algebra of functions on a semi-group (a group), and thus in a way of a semi-group (a group) itself, is a bialqebra (a Hopf algebra). Recently, quantum groups appeared in several fields of non-commutative mathematics; see e.g. [25,53,94]. It should be stressed that quantum groups are defined in different ways in the literature but they are Hopf algebras in all cases. We are concerned with the role bialgebras and Hopf algebras play in quantum probability as the 'non-commutative state space' on which our non-commutative white noise is modelled. Since we work with generalizations of probability measures, there is also a need for a positivity structure, and this is how *bialgebras and *-Hopf algebras corne in. These are bialgebras and Hopf algebras which have an involution compatible both with the algebra and the coalgebra structure. A *-Hopf algebra is close to what is called a matrix pseudo-group in [94]. The first time *-bialgebras appeared in quantum probability was in the paper [92] by W. von Waldenfels where the non-commutative coefficient algebra of the unitary group was introduced. After a crucial result on positivity "had been established [91,70]' L. Accardi proposed an algebraic framework for the general theory of quantum white noise on graded *-bialgebras. This program has been worked out in [5]. One of the main tools in quantum probability is the quantum stochastic calculus developed by R.L. Hudson and K.R. Parthasarathy in [41]; see also [15,16,51]. The connection between quantum white noise and quantum stochastic differential equations became apparent in [72]. Shortly after that, the question arose as to whether any white noise on a *-bialgebra can be realized as a solution of a quantum stochastic differential equation. Under the assumptions of boundedness and of Bose independence of the processes, an answer in the affirmative was given in [29]. Generalizing the results of [75], in these notes we treat the case of arbitrary unbounded processes with 'twisted' independence. We show that a white noise can always be realized on Bose Fock space. Using H. Maassens kernel method [51], we give an explicit formula for the processes. Indeed, the processes are solutions of quantum stochastic differential equations. The form of the equation is governed by the coalgebra structure of the underlying twisted *-bialgebra. Recently, there has been an attempt to generalize both the concepts of convolution semi-groups of states on a *-bialgebra (which are closely related to quantum white noise)

2

and of convolution semi-groups of instruments on a group [11] to a theory of convolution semi-groups of positive operator valued maps on a. *-bialgebra; see [12].

* The following discussion of four rela.ted topics in classical probability theory on groups and in quantum probability is intended to serve as a more detailed introduction to the mathematical problems we are concerned with. Con.ooluiioti semi-groups of probability measures. Let X t : n - G be a stochastic process indexed by time i ;::: 0 taking values in some (topological) group G. Suppose that the process X t is a white noise so that it has independent and stationary increments

i.e. the random variables

are independent for all choices of n E Nand t l < .. , < t n +! and the distribution of X. t only depends on the difference t - s. Then X t is determined up to stochastic equivalence by its I-dimensional distributions 'Pt which form a I-parameter semi-group of probability measures on G with respect to convolution. If 'Pt is weakly continuous at 0 we can differentiate 'Pt (f) for an appropriate class of complex-valued functions f on G to obtain the generator of 'Pt which in all cases of interest again will determine 'Pt and therefore Xt. For a Lie group G Hunt's formula [44] gives a description of all the generators of white noise in terms ofleft invariant derivations of first and second order and an integral which represents the' Poisson part' of XI' For a compact or locally compact abelian group the generator can be defined on the algebra R( G) of representative functions (see [34]) of G and the generators of white noise turn out to be the hermitian, conditionally positive linear functionals on R( G); cf. [35]. The principle of correspondence between I-parameter semi-groups of positive linear functionals and hermitian, conditionally positive linear functionals goes back to 1. Schoenberg [69] and will be called Schoenberg correspondence; d. [17]. Stochastic semi-groups. Let (X. t ) be a stochastic process indexed by pairs (s, t) of real numbers with 0 s t, taking values in the space Mn of complex n x n-matrices. Suppose that X. t satisfies the evolution equations X .. X. I

= X,I

Xtt = 1

almost everywhere for all r s t and that the 'increments' X. I are independent. A process X. t with these properties is usually called a stochastic semi-group; see e.g. [81]. Let the increments X. t also be stationary and assume that X. t converges to 1 in probability for t 1 o. (In this case, X. t is actually the increment process associated with a classical white noise on the general linear group.) Then there exists a stochastic process (Fdt2: o on the same probability space taking values in M; such that the following holds. The additive increments F. t = F t - F. are independent and stationary, F t converges to

3

o in probability for t equation

10

and X,t is the solution of the operator stochastic differential

the state on

A given

by

n.

, ••• ,

t n E T, bl , . . . , bTl E B},

Then the mappings

form a quantum stochastic process over (A, eI» which is equivalent to the original process (jdtET on (A, H(2), H(i)

the completion of

]((i),

such that

and U(7f(l)

for all t E T,

s e B and

all

E

0

]((1);

=

('71"(2) 0

see [4].

1.3. Independence. We introduce a notion of independence for n-tuples (it, ... , jn), n E N, of quantum random variables over the same quantum probability space (A, . whereas Fermi independence is the case when 0'0 = id and a1 is the parity operator on the 1'2­gradecl *­algebra E. Equation (1.3.3) is our starting point. Ot.her conditions will follow in a natural way. For instance, the multiplicativity of jk gives

ik(b 1b2)jl(C)

= jk(bdjl(O'd(b 2)c)jk(b 2) = jl(ad(b,)ad(b 2)c)jd b1 b2) .

.

which, on the other hand, equals

Thus it is natural to postulate

and, if we extend a linearly to Cl., we have that 13 is a module vector space. Similarly, the fact that jk is a *­map yields that 13 is a module *­vector space. Notice that for k> I so that t.here are commutation rules between all pairs (k, I) with k i- 1 (which one would like to have anyway). Next the *­algebra homomorphism property of jl yields that a), should be in Aut 13 which means that 13 actually is a module *­algebra. Notice that a : l ­­­­> Aut 13 is a *­group homomorphism if we equip Aut 13 with the *­group structure a >--> 0'* where a * () a =0' ­1 ( a *)* . Finally, we have by associativit.y

jk,(b1)jk2(b2)jk3(b3) = jk 3 ( ( d( db, .b2)dbl ).b3)jk 2(db, .b2)jk, (bd =

i»,((db, db2).b3)jk 2(db, .b2)jk, (bd

and this gives

d(>..b)>. = >'d b and 13 is a double­module *­algebra. Our considerations lead to the following definition. let 13 be a double­module *­algebra. An n­tuple (j1, .. . ,jn) of quantum random variables on 13 over a quantum probability space (A, 'I» is called left independent or (­y, a)­independent if the factorization property (1.3.1) holds and if (1.3.4)

19

for all b, c E B with b homogeneous and for all k

< I.

This also can be written

If (I.:U) is replaced by we speak of right independence or (0:" )-indepcndcnce. If nothing else is said we mean left independence. Notice that, in general, the order of jl,"" j" is important whereas for Bose and Fermi independence it is not. This is why, in the Bose/Fermi case, we have a notion of independence for sets of random variables rather than tuples. More generally, a family (j;)iEI of quantum random variables indexed by a set I with a partial ordering ::S is called independent if for all choices of n E Nand i 1 -« ... -« i-: in I the n-tuple (ji" ... , ji n ) is independent. 1.4. Commutation factors. We introduce an important class of double-module *-algebras, namely *-algebras with two compatible graduations and with the module structure given by a 'commutation factor'. For two abelian groups land M (written multiplicatively) a l-M-commutation factor . . IS a mapplllg q : l x M -+ C \ {o} = Co such that

q(A, 1'11'2) = q(A, ,dq(A, '2) q(AU\2") = q(Al' 1')q(A2,,)

(1.4.1) (1.4.2)

for all A, AI, A2 Eland 1',1'1,1'2 E Mj see [18]. If land Mare *-groups we add the condition

(1.4.3) Given an l-M-commutation factor q we form the sets = PEL:

and Then

q(A, Il) = 1 for all Il E M}

= {Il EM: q(A, Il) = and

1 for all A E L}.

are involutive subgroups of land M respectively and we form

the factor groups lq = and Mq = Since q gives rise to a lq-M qcommutation factor we always can pass to a commutation factor satisfying

q(.\, Il) = 1 for all Il EM=> A = e q(A,Il) = 1 for all A E l => Il = e The following choice of land M will be the most important for USj it is the case when the commutation rules are given by a complex number not equal to O. We take l = (Z x Z)twist and put M = t>. Notice that l is nothing but the free abelian

20 involut.ive semi-group generated hy an indeterminate y if we identify (1,0) win. y and (0,1) with y*. 1.4.1.

PROPOSITION.

q

Let land M as above. TIle equations

q((1, 0),1); q((ml' 11L2), n) = q",,71 q"'2 71

establisll a l-Lscortespotidcucc between l-M-commutat.ion factors q and non-zero complex numbers (w1licll we again denote by q). PROOF:

We have

by conditions (1.4.3) and (1.4.4). Using also (1.4.5), we obtain

q((O, 1), 1)-1 = q((O, 1), -1) = q((l, 0)*,1*) = q((I, 0),1)

1

.(>

Some remarks on special values of q. Denote by v( q) the minimum of all numbers I in No = N \ {O} such that (q/lql/ = 1 if the set of these numbers is not empty. If it is empty we put v(q) = 00. Case 1: fql #- 1,v(q) < 00, 1/ odd. Then lq = l/{(kv, -kv) : k E Z}

Mq Case 2:

Iql #-

1, v(q)

= M.

< 00, v even. t.,

Then

V v = l/{(k-, -k-) : k E

2

2

Z}

M q =M. Case 3:

Iql

= 1. Then

t.,

= l/{(m,m+ kv): mEl}

(lv)inv

Mq = M/{kv: k E I} = (lv)inv. where loo is to be read as I. Case 4: None of the cases 1-3: Then lq = land M q = M as before. In particular, if q is real, we are in one of the cases 1-3. If Iql #- 1 we have

i , = l/{(k, -k) : k E Z} M q -- IL7linv •

lid

If q = + 1 we have that lq and M q are trivial. If q = -1 we have lq = M q = 12 so that we are in the 12-graded case. A remark on commutation factors for the case when IL and M are of the form Iv: The only possible involutions on Zv are the identity and the taking of inverses, and one has to distinguish the two cases that the involutions in land M are of the same or of different type. If the involutions are of the same type the commutation factors

21

are given by complex numbers of modulus 1, and if the involutions are of different type they are given by real numbers not equal to o. Roughly speaking, the need for two different involutions arises only if one wants to use commutation factors which are not of mod ulus 1. A vector space V with an l- and an M-grading is called an l-M-graded vector space if the gradings CV(.\)hEIL and (Y(/L))/LEM are compatible, that. is if the V(A) are M-graded and the Y(/L) are l-graded subspaces of V. For an l-M-commut.at.ioll factor q we define a module structure on V by setting (1.4.4) for b E Y(/L). yT is a double-module vector space. Similarly, there are l-M-graded *vector spaces, algebras, coalgebras and --algebras which together with a commutation factor yield examples of double-module --vector spaces, algebras, coalgebras and *algebras. For a l-M-commutation factor q an n-tuple (jl' ... ,in) of quantum random variables over the quantum probability space (A, L( C) is an algebra homomorphism where we consider L(C) as the algebra of linear operators on C with the multiplication given by composit.ion. Moreover, EoR("p) ="p. Clearly, R("p) leaves invariant each subcoalgebra of C and we can define the convolution exponential exp."p E C* of"p by setting

(exp*"p)(b) = (E 0 exp{R("p)})(b). Let V be a comodule vector space and let W be a module vector space. We define the twist. map Th.a) : V 0 W ----> W 0 V to be the linear mapping given by

v E V, wE W. In the commutation factor case we write Tq for Th.a). If V and Ware involutive we have that IV> V''W ........ T(- 1 ) ( W

'Y. a

* IV> IV> * d*v ,,v * ) =v * ''w.

defines an involution on the vector space V 0 W. If V and W both are double-module *vector spaces then V 0 W is again a double-module *-vector space; the module structure is given by

.A.(v 0 w) = .A.v 0 .A.w. We denote it by V0h.a) W. A coalgebra (C, Ll, E) is called an involutive Cr, a)-coalgebra if it is a double-module coalgebra and if C is a double-module *-vector space such that Ll is a *-mapping from C to C 0h.a) C. Notice that, although the notion of a coalgebra is dual to the notion of an algebra, this relationship does not hold between *-coalgebras and graded *-algebrasj d. also the remark following the next proposition. This is also reflected in the fact that the dual algebra C* of a *-coalgebra, in general, cannot be turned into a *-algebra in the natural way. Indeed we have for "p, 'P E C*

and "p ........ "p* is an antilinear homomorphism but not an involution unless C* is commutative. We can form tensor products.

28

1.7.1.

PROPOSITION.

(i) For a, comotlule coelgebt« (C,D.,t) and a, module coslgebte (D,TI,7t') tlie triplet. (C0 'D,D. (:)h,u) n,£07t') witls

is a conlgebre: It is called ilic h, a)-tensor product of tlle coelgebres C and D and is denoted by C 0(-y ,0 ) 'D. (ii) For a couiodule algebra (A,M, 1\) and a module algebra (8, N, v) ibe triplet. (A (:) 8, M 0h,u) N, 1\ (0 II) Witll

M 0h,u) N = (M 0 N)

0

(id 0

0 id)

is all algebra. It is called the h, a)-tensor product of the algebras A and 8 and is denoted by A 0h,Q) 13, If A amI 13 both are double-module e-elgebtes ih en A 0h,Q) 13 is egeu: a double-module *-algebra. PROOF:

Repeated use ofthe properties of module and comodule algebras and coalgebras.

Similarly, (0', ')')-tensor products can be defined. In general, there is no analogue of the statement in (ii) on involutions for *-coalgehras. The h, a)-tensor product of two coalgebras which are also double-module *-coalgebras is not necessarily again a double-module *-coalgebra. However, we have ((D. 0h,Q) TI)(c 0 d))* = (D.

0(a,)')

TI)((c 0 d)*)

where the involution on the left hand side it is the one in (C0h,Q)D)0(-y,a) ( C 0 h ,a ) D ) and on the right hand side is the one in C 0h,Q) D. The space L(V) with V a double-module vector space is an example of a doublemodule algebra. The module structure is given by

(A.R)(v) = A.R(v.A). For double-module vector spaces V and W we have that L(V, W) and V 0 Ware again double-module vector spaces in a natural way. If VI, V2 , WI and W 2 are double-module vector spaces an embedding qi of L(VI , V2 ) 0 L(WI , W 2 ) into the double-module vector space L(VI 0 WI, V2 0 W 2 ) is given by

qi(R

(2')

T)(v 0 w) = Rv 0 (T.dv)w

for R E L(VI , V2 ) , T E L(WI , W 2 ) , v E VI and w E WI. In this way, L(V) 0 L(W) is considered as a double-moduled subalgebra of L(V 0 W) for double-module vector spaces V and W. This double-module algebra structure on L(V)0 L(W) coincides with L(V) 0h,a) L(W). The following can easily be verified.

29

1.7.2.

(i)

PROPOSITION.

a double-module *-veetor space V tIle twist map T("(,

T(V)

(2)(-y,c -v, l' E V. In the commutation factor case, we sometimes write T(V; q) for this structure to indicate the dependence of the comultiplication on q. In the case when IL = M = 71 2, the 12-graded *-Hopf algebra T(V) is also cocommutative. 3. ENVELOPING HOPF ALGEBRAS: Let g be a complex 12-graded Lie algebra. The universal enveloping algebra Env(g) of g is the tensor algebra T(g) divided by the ideal in T(g) generated by the elements

v0 w -

(_l)d(v)d(w l w (2) w

- [v, w],

v, w E g. One checks that this ideal is a graded coideal of the 12-graded bialgebra T(g). Thus Env(g) becomes a 71 2-graded (cocommutative) Hopf algebra. If g has an involution, that is g is an involutive vector space such that

[v, w]* = [w*, v*] for all v, w E g, then Env(g) is a 12-graded *-Hopf algebra. The 12-graded Lie algebras obtained from 12-graded *-algebras are examples of graded Lie --algebras. The complexification giC = g EB ig of a real graded Lie algebra g with v* = -v for v Egis another example. 4. COEFFICIENT ALGEBRAS OF MATRIX SEMI-GROUPS: In the beginning of the foregoing section we introduced the algebra R( G) of representative functions of a semi-group' G as a motivating example of a *-bialgebra. Roughly speaking, R( G) is the dual of the semi-group algebra in the sense that what was multiplication in R( G) is comultiplication in CG and vice versa. If G is a topological semi-group we denote by R o( G) the sub-s-bialgebra of R(G) formed by the continuous representative functions. If G is a group then R(G) and Ro(G) are *-Hopf algebras with the antipode (Sf)(x) = f(X-I). For a symmetric *-bialgebra B we denote by A(B) the set of all --algebre homomorphisms from B to C. Then A(B) is a convolution sub-semi-group of B*, and if B is a *-Hopf algebra, A(B) is a group with the inverse of X E A(B) given by X-l(b) = (X 0 S)(b). In the case when the group G is locally compact abelian or compact we have A(Ro(G» G which is the algebraic part of the Pontrjagin and the Krein-Tanaka duality theory respectively; see e.g. [34].

32 I II partie ular , if G is the grou P U(n) of II nitary n x n- matrices one can show that R o(G) is isomorphic to the commutative polynomial algebra M[n] in n 2 indeterminates ;"'f.ltl, x kl' k, I 1, ... ,n, divided by the ideal I in M[n] generated by the elements

(1.8.1 ) see e.g. PO]. The involution is given by co unit are obtained by first extending tl : M[n]

-t

with

(Xkz)*

xkl and the comultiplication and

=

M[n] 0 M[nl; e : M[n]

--->

C

n

tlxkl

=

L

Xkm

0

Xntl;

(Xu

=

(1.8.2)

bkl

In=l

to a *-algebra homomorphism, turning M[n] into a symmetric *-bialgebra, and then showing that the ideal I is a *-coideal. The antipode of Urn] = Ro(U(n)) = M[n]/I is given by Xkl >--> x 1k . A sub-s-Hopf algebra of U[2] is SU[2] = R o(SU(2» which is obtained by dividing U[2] by the ideal generated by the elements xiI - Xn and xi2 - X21' Another commutative symmetric *-Hopf algebra is defined as follows. We denote by H[2] the commutative *-algebra generated by the indeterminates x,y,y-l,z and Z-1 with the relations yy-l = zz-1 = 1. We turn H[2] into a symmetric *-bialgebra by defining z and y to be group-like and z to be (z, y)-primitive. We also are interested in the quotient H(O)[2] of H[2] by the *-ideal generated by the element z -1. Then H(O)[2] is the commutative *-algebra generated by z , y and y-l with the relation yy-l = 1 and such that y is group-like and x is (1, y)-primitive. Finally, H(OO) [2] is defined as the sub-v-bialgebra of H(O) [2] generated by x and y. An antipode of H[2] is given by x >--> - Z -lxy- 1, z" >--> -(z*)-: 1 x* (y* )-1, so that H[2] and H(O) [2] are *-Hopf algebras. We have that A(H[2]) is the Heisenberg group H(2) formed by the 2 x 2-matrices of the form

13

a, 1 E Co,

E C

of H(2) formed by the matrices

Moreover, A(H(O) [2]) is the subgroup H(O)(2) 1 E Co,

13

E C, and A(H(OO) [2]) is the semiWe say that Urn]'

group H(OO)(2) of all complex 2 x 2-matrices of the form

SU[2]' H[2], H(O)[2] and H(ooJ[2] are coefficient algebras of U(n), SU(2), H(2), H(O)(2) and H(OO)(2) respectively. 5.

NON-COMMUTATIVE ANALOGUES

AND DEFORMATIONS

OF COEFFICIENT ALGE-

The non-commutative analogue of M[n] is the *-algebra M(n) generated by non-commuting indeterminates Xkl, k, l = 1, ... , n, which was already introduced in Section 1.6. We have seen that, for a given commutation factor q E Co, the *-algebra M(n) can be turned into a (1 X 1)twist_1 inV_grad ed *-algebra. If we define

BRAS:

tl : M(n)

--->

M(n) 0 q M(n);

E:

M(n)

--->

C

33

again by extension of (1.8.2) we obtain an involuti ve q-hialgebra which we denote by M(nj q). Next we have

1.8.1. PROPOSITION. Let Iql = 1. Tlien tile two-sided ideal I in M(n; q) generated by tile elemeu is of tite form (1.8.1) is a lq-Mq-graded » A is a *-algebra homomorphism. The proof of (i) is complete if we show that i-, * jot and jTt agree on the generators v of T(V). But t.his is clear, because we have PROOF:

i-, * jot:

(ii) and (iii) are proved in the same manner.\> A linear functional je on an involutive q-bialgebra B is called conditionally positive if 1jJ(b* b)

2' 0 for all bE B with c(b)

=

O.

1.9.2. THEOREM. Let (j,c)(o,t)ETL be a white noise on tIle involutive h, o::)-bialgebra B. Then the I-dimensional distributions 'Pt form a I-peremeter convolution semi-group of states and there exists a unique linear funetional1jJ on B such that (1.9.1)

'Pt = exp,,(t1jJ). Moreover, 1jJ(b)

=

d dt 'Pt(b)!t=o for all b E B

and 1jJ is hermitian, conditionally positive and a-invariant and seiilies 1jJ(1) PROOF:

(1.9.2)

= 1.

We have 'Po

* 'Pt = 'P. * 'Po,o+t = ( 0 e.

40

which by Hie commu tation rilles is a *-algebra homomorphism. The increment property yields )eT,T 0 jeT = jT and the factorisation property gives 1> 0 jeT ,= 'PeT' By the property of an inductive limit it follows that there exists a unique *-algebra homomorphism (-) such that jeT = (-10 JeT and = 0 e.(>

N ow we have the reconstruction theorem. 1.9.7. COROLLARY. Let {'Pt : i E be a l-para,meter convolution senu-gxoup of a-inverient states on an involui.ive (" a)-bialgebra B, and suppose that lim 'Pd b) =, f( b) [or all b E B. ttG Then there is a white noise ou B wiil: I-duneusiouel distributions equal to 'Pt. Moreover, two wliite noises on B are equivalent if and only i[ their l-dimeusionel distributions, and thus their generators, egree.t» In Section 2.2 we will see that any conditionally positive linear functional appears as the generator of a white noise.

2. Symmetric white noise OIl Bose Fock space. This chapter is t.he heart of the theory. We show that. an arbitrary symmetric white noise can be realized as a process of liucar operators on some suitable Bose Fock space. The operators will, in general, he far from bounded, but they always will be defined on a common invariant dense domain in the underlying Fock space. Our operators are kernel operators in the sense of H. Maassen; see [51]. The original work of Maassen is generalized by considering kernels with three arguments which take values in spaces of operators on Hilbert spaces. This generalizatioIl is necessary to include dif.Terenlial second qu.aniizaiion. processes on the one hand and Bose Fock spaces over spaces of square-integrable fund-ions on IR+ taking values in an arbiiraru Hilbert space on the other hand. There has already been done a considerable amount of work in this direction by P.A. Meyer, M. Lindsay and K. R. Parthasarathy. In particular, we widely made use of [49,50,57]. We arranged the preparatory part of this chapter always in view of our main theorem which is the representation of white noise on Fock space. 2.1. Bose Fock space over L 2(1R+,H). Let (A 1 , d ,... ,(An ,

V is called a (p, c:)-l-cocycle if for all a, s « A

TJ(ab)

= p(a)I'}(b) + 7)(a)c(b).

Let V be a pre-Hilbert space and assume that p(A) C H(V). An antilinear mapping fJ : A ----> V is called a (c:, p)-l-cocycle if for all a, bE A t9(ab) = c:( a )19( b)

+ p(b)*fJ( a).

For a linear functional 1jJ on A the [e, e)-coboundary of 1jJ is the bilinear form 81jJ on A given by

81jJ(a, b) = c:(a)1jJ(b) -1jJ(ab) + 1jJ(a)c(b).

A proof of the following will be given in Section 2.5. 2.3.3.

THEOREM.

Let 13 be a bialgebra and assume that p(13) C H(D), and that

is a representation of 13 is a (p,c)-l-cocycle fJ is a (c,p)-l-cocycle 81jJ(a, b) = -(fJ(a),7)(b)). p

7)

Then k,t is a representation of 13 on ED for all (s, i) E T. We now come to a construction which is central for our theory. Let A be a *-algebra and let c be a *-algebra homomorphism from A to C. Moreover, let 1jJ be a hermitian, conditionally positive linear functional on A and assume that 1jJ( 1) = O. We form the positive sesquilinear form

(a, b),p = 1jJ((a - c:(a)l)*(b - c:(b)l))

56

on A and divide by the null-space =

{a C: A: (a, a)".

=

O}

of this form to obtain a pre-Hilbert space D". = AjNrf;' Denote by canonical mapping. Then the following theorem holds.

:A

-->

Drf; the

2.3.4. THEOREM. T11e eqtuiiiou (2.3.4) defines a e-represeuietiou of A all

Moreover, 17rf; is a (p, c)-l-cocyc1e and

PROOF: We prove that (2.3.4) defines a *-representation. We have

= 111N(a(b - c(b)l))11 = 7{'((b - t(b)l)*a*a(b

=

t(b)l))

- t(b)l)))

S 111J,p(b)IIIIr7,p(a*a(b - c(b)l))11 by the Cauchy-Schwartz inequality which shows that p,p(a) is well defined. Since aib c( b)l) EKern c it follows that p", is a representation of B on D,p. Moreover,

(prf;(ahrf;(b),1J,p(c))

>

= 1/;((b - c(b)l)*a*(c - c(c)l)) = ('ry,;,(b),p¢,(a*hrf;(c))

which shows that

is a *-mapping from A to H(D",).(>

Now we can state the 2.3.5.

REPRESENTATION THEOREM FOR SYMMETRIC WHITE NOISE. Let B be a *-

bia1gebra and assume that p is a e-represeutetion of B on D . TJ is a (p,t)-l-cocyc1e . 81/;(a, b) = -(TJ(a*), TJ(b)).

If we pu t {) = ij, then, in the vacuum state, (k. t )(s,t)ET form a symmetric w11ite noise on B with generator 1/;. Conversely, a symmetric white noise on B with generator .,p is equivalent to (k s t ) with p = P,p and TJ = TJ,;,. PROOF: We have

ij(ab)

= TJ(b*a*) = p(b)*ij(a) + ij(b)c(a)

which shows that {) is a (c,p)-l-cocycle. Combining Prop. 2.3.2 (iv) and Theorem 2.3.3 we obtain that k. t is a *-homomorphism. Prop. 2.3.2 (i) is the increment property and 2.3.2 (ii) gives the independence of increments in the vacuum state. The stationarity of increments and the weak continuity follow from Prop. 2.3.2 (iii). Thus k. t form a Bose white noise on B and, again by 2.3.2 (iii), we know that 1/; is its generator. The remaining part follows from Theorem 2.3.4.(>

57

2.4. Quantum stochastic integrals and quantum Ito's formula. The aim is to establish a quantum stochastic integration for 'non-anticipating' operator-valued processes on Fock space. But what are the integrators? Led by the realization of Brownian motion as a sum of annihilation and creation operator processes, two quantum stochastic Ito integrals, against the 'separated' processes of an nihilation and creation operators, were introduced by R.L. Hudson and K.R. Parthasarat.hy [41]. However, apart from the usual Riemann-Bochner integral for operator processes, also another type of integra.tors was used in [41], namely processes coming Irom the differential second quantization procedure. It will become clear later that, for our purposes, we actually need all four types of integrat.ors. Our t.heory itself will provide a justification of the choice of integrators. They will t.urn out. to be exactly the components of an arbitrary additive white noise; see Section 4.2. Following If. Maassen, we introduce t.he integrals for the kernels first. Consider the creation process ED. It will leave [.D invariant and we have

for a fixed element.

where A t ( ses are

is the annihilation process on ED' The corresponding kernel-valued proces*

at(O(cr,T,/?) = and

{ a*(t)(O

0

a(t)(O

at(O(cr, T, /?) = { 0

if

T

= g = 0, a = {t}

otherwise

ifcr=T=0,/?={t} otherwise

respectively, i.e. Q.;(O = A;(O and f!t(O = At(O. In view of the basic formula for the definition of t.he third kernel-valued process

A(t)(T) At(T)(cr,T,I?)= { 0

if a = I? = 0, T = {t} otherwise

for T E L(D) seems only natural, We set. At(T) = }t (T). If T E H(D) we have At(T) E H(E D ) and At(T)* = At(T*). It is clear that At(T) is nothing but the differential second quantization of the operator f f--+ A[o,t)Tf on H) where X[O,t) is identified wit.h the corresponding multiplication operator on H) (it cuts f from time t onwards) and T f is to be understood pointwise. The additive increments of all the three types of processes are adapted or non-anticipating in the following sense. For a measurable subset B of IR+ the trace B n S of B in S is the subset {B n w : w E S} of S. The mappmg

given by

58

is unitary which follows if we look at exponential vectors. We identify

and F(£2(1ll 1 ,TT)) via Un- In particular, for F; = F( [;2([s, t), H» we have

to which corresponds a decomposition

of [D. A process (s,t)

f-t

k. t E L([D) is called adapted if there is a process

such that. k. t is the ampliation id 0 simply means that

kst(a-, T, g)

= 0 a.e ,

0 id of unless a-

On the level of the kernels this

U

TUg C [s, t).

We denote by AD the class of all families (k.d(s,t)ET of kernels in /CD such that

(AI) (k.d is adapted (A2) the vector space E 2 of (El) can be chosen the same for all k. t if t stays in some bounded subset of III+ , and the measurability condition (K2) holds if we also vary (s,t) in T C Ill+ X Ill+ (A3) for all kst one can choose the same constant c of (K3) if t stays in some bounded subset of IR+. By Prop. 2.3.1 the kst(b) associated with a triplet (1/, p,{), 1/J) on a coalgebra Care examples of processes in AD. The set AD becomes a vector space with the obvious structure. Moreover, the product of two elements in AD is again in AD, and AD becomes an algebra. The set AD,h of elements (k.d in AD with the additional property that k. t E /CD,h for all (s, t) E T is a *-algebra with = «k.d*). As a motivation for the definition of the quantum stochastic integral of (k. t ) E AD against dA;, dA t and dA t consider a simple process if "i

:s: t < Tj+l

and a- U TUg C [s, t)

otherwise

(2.4.1)

in AD where TO = s < Tl < ... < Tn+! = t and where kj(a-,T,g) E /CD. Then the following holds for the corresponding 'Ito sums' n

(L kSTj(a;Hl (0 -

T, g)

j=O n

= Lkj(a-n[s,Tj),Tn[s,Tj),gn[s,Tj»X j=O

59

a; --a;.

where we put 0;. = We assume that the partition TO < ... < T n+1 of [11, t) is finer than the one given by a U TUg ,1 [s, t) which we can do because for any given partition TO < ... < 'I'n+1 we have a representation of k.t(cr, T, e) in the form (2.4.1). Then the above sum is equal to 0 unless a I 0, a = {i l < ... < in}, S ::; iI, S ::; r < in < t for all T E TUg. In this case, it is equal to

where j is such that

t
'b(1)5(b(2))d(b)-1 >..-1

(id 0 5-r) 0 s: )(>.b) =

= f(b)l

= f-r('\b)l

72

where we used that Then

f

is o-invariant. Suppose that B is an involutive (r, n)-Hopf algebra.

=

S.,(S'1(PW)*)

1

S'1((P* )-lS(b* )d(b*t )*)

= 5'1 (d(bt 1S(b*)* A-I) = /\S(S(b*)*)d(b)d(b)-l = Ab which proves that Bh,a) is an involutive Hopf algebra.O The relationship between a (", o:)-bialgebra B and its symmetrization Bh,a) can be expressed by saying that the diagrams III

B

Bh,a)

-------->

1

(Ill GidGid)OL'f

B0B

and

'IL

8

,

8 h,a) 0 8 h,a)

Bh,a)

1s ,

80B

'IL G'IL (

Bh,a) 0 Bh,a)

commute. There seems to be no reason why we should prefer (r, 0:) rather than (0:, -v]. Indeed, one also can consider the tensor product 8 0(a,'1) eM and the analogues of Prop. 3.1.1 and Prop. 3.1.2 hold. The choice of (r, 0:) is adjusted to left white noise. However, if B is an involutive (0:, r)-bialgebra we can form the symmetric *-bialgebra 8(a,'1) = «BOP)h,a))OP. It is immediate to check that the comultiplication f);.'1 of 8(a,'1) is given by for b E B. In the commutation factor case, we write B q and Bq for Bh,a) and B(er,'1) respectively. At the end of this chapter we will see that, while Bh,a) is the suitable symmetrization for left white noise, B(a,'1) is the natural symmetrization for right white noise.

We form the symmetrizations for some of the examples of 1.8. Let V be a double-module *-vector space. The comultiplication for the symmetrization T(V)h,a) of the involutive (", o:)-bialgebra T(V) is given by f);.'1 v = 1 0 v + v 0 d(v )

for v E V, i.e. v is (1, d(v))-primitive. Moreover, an antipode of T(V)h,a) is given by S(v) = ':"-vd(v)-I.

73 We turn to our special example when \l is spanned by x and z " and we are in the commutation factor case, q E Co, with the l-M-graduation given by IL = (1' x 1')twist, M = 7l.in v. If one specializes to a lq-Mq-graduation one has to add certain relations for y and y'. The *-Hopf algebra T(\l)q can be described as the *-algebra generated by x and indeterminates y, y-1 with the relations y* x = qxy*

yx == qxy,

[y, y "l = 0 ,

yy -1

=

y -1 y

= 1,

and where y is assumed to be group-like and x (1, y)-primit.ive. This is exactly the deformation

of the subgroup H(O)(2) of the Heisenberg group. Going one step

further, we form the associated *-Hopf algebra of the deformation of H(O)(2) which turns out to be equal to the deformation H 1;q,a(2) of the Heisenberg group H(2). In fact, this is analogous to the commutative situation. Let us look at the commutative and symmetric tensor *-Hopf algebra C[x, x*] = which we regard as an involutive q-Hopf algebra with q = 1 and l = (71. X 7l.)lwisl, M = 71. in v • If we apply our construction we obtain (qx, x*])q = H(O)[2], and, applying it once more but in the other direction, to H(0)[2] we obtain H[2] so that H[2] = ((qx,x*])q)q. The associated *-bialgebra (M(n; q))q is the *-algebra generated by x u, k, l = 1, ... I n, and y,v:' with the relations YXk/ [y, yO]

= qXklY, =

0,

Y*Xkl

yy-1

= qXklY*

= v:» =

1,

and such that y and y-l are group-like and

L n

ll-yXkl =

Xkm 0 yk-m Xml.

m=l

The next proposition will yield a simple relationship between multiplicative q-white noise and multiplicative symmetric white noise. 3.1.3. PROPOSITION. Tbe *-algebra bomomorphism

e : M(n; 1) ---> (M(nj q))q given by is a coalgebra embedding. Moreover, e respects the unitarity relations (1.8.1), so that bialgebra embedding ofU(n; 1) into (U(n;q))q.

e

gIVes nse to a *-

74 PROOF':

We have 0)(Xkd

0

= (y-k

n

(9

y-k)( L

Xkm

(9

yk-mxmd

m",l n

=

""'

Y -k Xkm

",-m 'd Y

xml

nJ,=l

Next we have n

n

0( L

XkmX;m) = L

ru c- L

ykXkmX;myl

m=l

n

= y-k(L XkmX;m -

kI1)yl.

rn.zr I

3.2. Schoenberg correspondence. The following result goes back to Schoenberg [69]; cf. also [17,63]. Let X be a set and L a kernel on X, i.e. L is a complex-valued function on X x X. A kernel L is said to be positive definite if Lz,zjL(Xi'X)) 2 0 ill

for all choices of n E Nand Xl, . . . ,X n E X, Zl, ••. ,Zn E C. Then the (pointwise) exponentials exp(tA) of a kernel A are positive definite for all t 2 0 if and only if A satisfies the conditions

A(x,y) = A(y,x)

for all z , y E X

and Lz,zjA(Xi,Xj)

2

0

(3.2.1 )

v.i

for all choices of n E N, Xl,"" X n E X and Zl, ••• , Zn E C with L:i z; = O. A kernel satisfying condition (3.2.1) is called conditionally positive. The correspondence between I-parameter semi-groups of positive-definite kernels and conditionally positive-definite kernels is often called Schoenberg correspondence. The principle of Schoenberg type correspondence plays an important role in mathematics. In this section, we treat a generalized Schoenberg correspondence where the pointwise exponentials are replaced by convolution exponentials with the convolution coming from a general coalgebra structure. Given a conditionally positive, hermitian linear functional e with 1jJ(1) = 0 on a symmetric *-bialgebra B we constructed a white noise on B with generator 1jJ in Chapter 2. Since 'Pt = exp*tljJ are the I-dimensional distributions of this white noise, it follows that 'Pt are states. This establishes a Schoenberg correspondence for I-parameter convolution semi-groups of states. We generalize this result in two directions. We consider positive

75

sesquilinear forms on coalgebras which generalize the concept of states on *-bialgebras. Then we also allow a non-trivial double-module structure to be present. Let V be a double-module vector space. We form the conjugate vector space V of II which as a set is {v : 11 E V}. The addition and the scalar multiplication are defined by

v + W = 11 + 'W, zv =

"iv,

11,1£1

E V, z E C.

A graduation of V is given by the family (VP')hEl' becomes a double-module vector space if we put

The graded vector space V

.x.v = v ..x*. Let V and W be two double-module vector spaces. An isomorphism 0 from L(V, W) to L(V, W) is given by

(0R)(v) = R(v),

R E L(V, W), v E V.

We identify the elements of L(V, W) and L(V, W). The conjugate vector space V of the tensor product is identified with V 0 W via the map v0

'W

I->

V0

@

W

w.d(v).

The vector space C is identified with C in the obvious way. One checks that the conjugate (C, E") of a double-module coalgebra E) again is a double-module coalgebra. 3.2.1. PROPOSITION. Let C be a double-module coalgebra. Then the (-y,a)-tensor product C0(,.,0) C of coelgebres is turned into an iuvolutive (-y, a)-coalgebra i[ we define the involution on C 0 C by

(c 0 d)* =

d0

c [or c, dEC.

PROOF: Clearly, with this involution, C 0 C becomes a double-module *-vector space. Moreover,

(E"@E)((c0d)*) = E(d)E(C) = E(C)E(d) which shows that E" 0 We have

E

is a *-map. It remains to show that

d(1) 0 C(l) 0 d(2).(d(d(l))d(c(1))) 0 C(2)

and 0(,.,0)

=

is a *-map, too.

0 d)*)

0(,.,0)

=L

0(,.,0)

0

d)

L C(l) 0 d(1) 0 (c(2).d( C(l))).d(d(l)) 0 d(2),

(3.2.2)

76 so that

((t.

=

0(-y,a)

L

t.)(c 0 d))*

d(1) @ c(1)

0

d(2).d( d(l)C(1))

0 (d( d(l)C(l) ),C(2) ).d(d(l)C(l»)

which equals the right hand side of (3.2.2). By regarding sesquilinear forms on a double-module coalgebra C as elements of L(C

°

C), we obtain the notion of the convolution product of sesquilinear forms on doublemodule coalgebras. A sesquilinear form L on a vector space is called positive if L( v, v) 2': 0 for all v E V. For a coalgebra C and b E C we denote by L b the sesquiliuear form on C with

Lb(c,d) = L(c - E(c)b,d - E(d)b). A sesquilinear form A on C is called conditionally positive if A(c,c) 2': 0

for all c E C with E(C) = O.

For a £'2-graded coalgebra the convolution exponentials exp..( tA) of an even sesquilinear form A on C are positive for all t 2': 0 if and only if A is hermitian and conditionally positive. The proof of this rests on the question under which additional conditions on the sesquilinear forms the convolution product of two positive sesquilinear forms again is positive. For q = 1 no extra condition is needed. For l = M = £'2 and q = -lone of the sesquilinear forms has to be even. In the general case we have 3.2.2. PROPOSITION. Let C be a double-module coalgebra. Then, under the condition that B is a-invariant or A is even, the convolution product A * B of two sesquilinear forms A and B is again positive. PROOF:

We have for C E C (3.2.3) i,j

which, in the case when A is even equals

L A(Cli 18> Clj )B(C2i 0 C2j) iii

which is

2':

0 because the Schur product of two positive definite matrices is again positive.

If B is a-invariant (3.2.3) equals

L

A( Cli 0

Clj )B( C2i .d(Cli) 18>

C2j .d(Clj ) -1)

iii

= LA(Cli 18> c l j )B (d (Cli ).C2i 0d(Clj).C2j) i,j

which is

2':

0 for the same reason.O

77

Now we reduce everything to the symmetric case. We equip C1 with the trivial doublemodule structure. Then (C1 ) is identified with = CL A*0c.A* and

C1

0 C1 is mapped into

(C The following result is now an immediate corollary to the preceding proposition and to Theorem 3.1 of [70]. 3.2.5. THEOREM. Let A be an a-invariant sesquilitieer form on a double-module coalgebr« C. We consider the following statements: (i) A is conditionally positive (ii) there is an element c in C with fCC) = 1 such that A C is positive (iii) AC is positive for all c E C witll fCC) = 1 (iv) A is hermitian and conditionally positive (v) exp.. (tA) is positive for all t :::: o. Then the implications

(i) 11

(iv)

¢:?

(ii)

¢:?

(v)

¢:?

(iii)

hold.(> Next we have the following proposition. 3.2.6. PROPOSITION. Let B be an involutive(r,a)-bialgebra. Then the linear mapping

given by

K(bl2> c) = b*c is an even coalgebra epimorphism. Moreover, K is a-invariant, and K* ( b(;).(d(b(l»*d(c(l)) 0 C(2)

L b(l) * C(l) 0 b(2) * .( d( b(1»*d(C(l) )C(2)

= Ll(b*c)

=(LloK)(b0c)

79 which shows that K is a coalgebra homomorphism. Since K(e 0 b) = b, the map K is surjective. For a linear functional ({J on B

K*«({J)(b 0c) = ({J(b*c) and it is immediate that K*( ({J) is hermitian (positive, conditionally positive) if and only if ({J is.(jttt,(b l

=

) ...

.

Now let. i1

< ... < t n +1 and bl , ... , bn

E

B.

jtnt n+ . (bn ))

1>(ittt, ·h.a)(b 1 )It,L .b.a)(d(b )) ... ·b.a) (b ) .(-y,a) (d(b ))) 1 itnt n ltn+t L n n+t

= exp*( (t2

t 1 h'-'a)( bdexp*( (t 3 - t z )y,a)( d( bl )b2 ) ..• .. . exp.((tn+1 - tn)y,a)(d(bd··· d(bn-dbn) exp*(L - t n+1 )y,a )(d(bd .. . d(bn_dd(b n)) -

= 'Pt,-t.(bd·· .'Ptn+t-tn(bn

)

which shows that (j.t) has the factorization property for time-ordered correlations. However, since y, is assumed to be a-invariant, the white noise factorization property (1.3.1) holds. This gives the independence of increments (WN2). (WN3) and (WN4) follow by construction. This proves that (j.d is a white noise on B. Its infinitesimal generator is y,. The restriction L




can be omitted, because

will always make sense in

the special realization of which we will describe in the next chapter. Suppose that y, is a generator on the involutive (a, l' )-bialgebra B. We form BOP and (BoP)c.r,a), and Y,a asa generator on (BOpkr,a)' Let be a white noise on (BOPkr,a) with generator ./, We put J.(a.. r ) = (hb,a))op = h(-y,a) Then 'l'a' .t .t L-t,L-.·

. (b) --]0. .(a'1')(d b ) ].t .(a'1')(b)

].t

(3.3.1 )

gives a white noise on B with generator y,. There is a more general way to obtain a realization of (a,I')-white noise. We know that is a (symmetric) white noise on B(a,1') = «BOP)(-y,a))OP. Thus, if we consider Y,a as a generator on B(a,1')' then for any realization of a white noise on B(a,1') with generator Y,a we have that (3.3.1) is a realization of a white noise on B with generator y,.

4. White noise OlL Bose Fork space. Using the symmetrization principle and the fact that group-like clements give rise to second quantization processes, we can immediately apply the results of Chapter 2 to the case of a general Cr, a)-bialgebra. Additive twisted white noise is an integral over the second quantization process of the action Q with respect to a sum of creation, preservation, annihilation and scalar processes. After describing additive noise, we observe that, in complete analogy to the symmetric case, we can write a twisted white noise as the solution of a quantum stochastic differential equation against additive twisted white noise. Bose and Fermi quantum Brownian motion [22,6,7] and the 'quantum Poisson process' are special cases of additive white noise. The Azerna martingales [10,26,61] are closely related to additive q-white noise and to interpolations between Bose and Fermi quantum Brownian motion; for the latter see also [50]. In the case of multiplicative white noise, we obtain a particularly simple form of the quantum stochastic differential equation: it can be written as a linear quantum stochastic different.ial equation on the tensor product of the Fock space with some finite-dimensional "initial space'. We obtain a characterizat.ion of unitary white noises which turn out to be unitary evolutions in the sense of [41,42] on en 12> F(L 2(1R+, H)) where H can be an infinite-dimensional Hilbert space. In the case of a 1'2-graded cocomrnutative *-bialgebra white noise can be written down explicitly without using kernels or quantum stochastic calculus. This generalizes the results for group Hopf algebras and tensor Hopf algebras. We obtain a classification of the infinitely divisible representations [56,86] of 1'2-graded Lie *-algebras. 4.1. Group-like elements and realization of white noise. Since the elements of IL c B(-y,a) are group-like, in view of Theorem 3.3.1 we must investigate the behaviour of jst(b) for b E B, b group-like, B a symmetric *-bialgebra. Before we begin with a combinatorical lemma, we introduce some notation. In a similar way, it was already used in Chapter 2. For a vector space V and w = {t l < ... < tn} E Snl we put

VOW

=

Vi, 12> ••• 12> Vi n

with vt k a copy of V and we put V00 = C. For a linear mapping R : V vector space V to a vector spave W we let R0 w denote the mapping

-+

W from a

with R t k a copy of R and we put R00 = id. Furthermore, for linear mappings Rt" ... ,Rt= : V -+ Wand T = {t l < ... < ttn} C w we denote by

(Rt , 12> ••• 12> R, •.J (w)

: V0 w -+ WOw

the ampliation of R t , 12> ••• 12> R t = to Vow. 4.1.1.LEMMA. Let V be a vector space, C a coalgebra and p : C

-+

L(V) linear. Then

the equality

(pOW

0

= l)(p T(W

fid)0 T a

(4.1.1)

82 holds for all

w

E S.

PROOf': (hy induction on n = #w) Equality (1.1.1) is correct for n = O. Suppose it holds for n. Using the coassociativity and the counit property, we obtain for w = CTU{t}, #CT = Tt, t > 0-, that the right hand side of (4.1.1) equals

- ((p - Eid)0 T 0 e id] 0 .6. TU{t} (b)(w»)

= (pO" 0 .6.,,(b)) 0

id

+ ((p0

CT

0.6.(7) 0 p) 0 .6.(b)

- ((p0 0.6.,,) 0 Eid) o.6.(b) CT

= (pO" 0 p) 0 (.6." 0

id) 0 .6.(b)

which is the left hand side of (4.1.1). In particular.

r .t(R)

is the unique solution of

and

(r.t(R)F)(w)

=

R®w F(w) for w C [s,t).

The structure of white noise on *-semi.group bialgebras CG and, therefore, on group *-Hopf algebras is now clear. The operators j.t(x), x E G, are given by kernel processes of the form (4.1.2) with () = ij and they map E(f) to (4.1.3). In the case of a group G, the j.t(x) are unitary and this is the well-known formula for j.t(x)E(f) that can be found in [32,63]. However, the applications of Prop. 4.1.2 are much wider.

Let 1/J be a generator on the involutive (-y,a)-bialgebra B and let be ilie realization on Bose Fock space of the white noise on Bh,a) with generator 1/Ja. Then for A E l the operators ,a)p.. ) form the second quantization process r.t(Pa(A)) of Pap.. ). Moreover, we have for b E Band), E l 4.1.3.

PROPOSITION.

Pa(A}1](b) = 1]()..b). PROOF: Since A is group-like and lla().) = 1]a(.\*) = 1/Ja(A) = 0 we deduce from Prop. 4.1.2 that

Next we have for b EKern c and A E l

II1la(A.b) - 1Ja(Ab)11 2 = 1/Ja((A.b - )'b)*(.A.b - Ab)) = 1/Ja((b* .A*)(.A.b)) + 1/Ja(A*(b* .A*)(A.b)A) -1/Ja((b* .,\* )(A.b)A) -1/Ja(A*(b*.A*)(A.b)) =

o.

where we used the relations in Bh,a) and the o-invariance of 1/J.(> In the following, we denote the *-representation A >--> Pa (A) of l on D again by the symbol a, so that 1)(>... b) = a),1J(b).

84 A remark onLhe commutation factor case under the assumption that M some group G: Since 1/1 is M-even, we have that D is the orthogonal sum

= c»:

for

r

of the linear su bspaces D(J1.) = 7/(8(J1.)) and a" 8(J1.) is simply multiplication by q(>",1-1). Now we are in a position to formulate the representation theorem for general (1',0:)white noise. 4.1.4.

THEOREM.

Let

1/J

be a generator on tIle iuvoluiive (')', a)-bialgebra 8. 'I'Iien tlle

quantum siocliestic backward integral equations

(4.1.5) with and

it = A;(1](b)) + At(p(b) - E:(b)id)

+ A t(1](b*)) + 1/J(b)t

have unique solutions j,t(b) in AD' In tIle vacuum state, the j,t form a white noise on 8 with generator ,p. Let be the unique solution in AD of the quantum stochastic forward integral equation

.h,a)(b) i.;

= E(b)id +

It (dA;(ry(d(b(l)).b(2))) + dA,(a(d(b(l)))p(b(2)) - E(b(2))id)

+ dA,(1](b(2))) + 1/J(b(2))dr).

(4.1.6)

Then (4.1.7) The realization on Bose Fock space of a symmetric white noise on with generator 1/Ja is the unique solution in AD of the equation

PROOF:

8h

,0 )

(4.1.8 ) Using the homomorphism property of Pa and the facts that Tfa is a 1-cocycle with respect to Pa and ,p is a-invariant, we see that is also the unique solution in AD of (4.1.6). By Theorem 3.3.1 and Prop. 4.1.3 we have that (4.1. 7) is a realization of white noise with generator 1/J. We multiply equation (4.1.8) by rt,oo(ad(b)) and use

85 to obtain (1.1.5). The uniqueness assertion in the first part of the theorem follows from the fact that, if J.t(b) is a solution of (4.1.5), then is a solution of (1.1.8). The latter has a unique solution in dv by Theorem There is a version of Theorem 4.1.1 [or (0','"'( )-white noise. Equation (4.1.5) is replaced by the forward equation (4.1.9) with

dIT(b) = fT(O'd(b»)diT(b), (4.1.6) is replaced by the backward equation .(O'")')(b) i..

= t(b)id +

1 t

L(dA;(1/(b(1»))

+ dA T(p(b(1»)O'd(b(2)

- t(b(1»)id)

(4.1.10)

+ dA T(1](d(b(2»)* .b(l))) + and (4.1.7) is replaced by

4.2. Primitive elements and additive white noise. We apply Theorem 4.1.4 to tensor-v-Hopf algebras and show that the equations (4.1.5), (4.1.6) and (4.1.9), (4.1.10) can be rewritten as simpler integral equations with integrals against additive left white noise and additive right white noise respectively. These simpler equations are the precise analogues of the symmetric case. Again we start from a general consideration.

4.2.1. PROPOSITION. Let B be a symmetric bialgebra, let y E 13 be group-like and b E B be (1, y)-primitive, and let (1], p, {), 1jJ) be as in Theorem 2.3.3. Then we have for

O"UTUI?

=

{t 1 < ... < tn} T,

=

I?)

(1jJ(b)1jJ(y)-1(exp((t 1 - s)1jJ(y)) -1)1'C(t1;0",T,I?)(y) + 1'C(t 1;0",T,I?)(b))

(4.2.1)

exp((t - td1jJ(Y))(1'C(t 2 ;0", T, I?)(Y)'" 1'C(t n ; 0", T, I?)(Y))' Moreover, for ist = k. t we have that (4.2.2) and i.t(b) is the unique solution in AD of (4.2.3)

86 PROOF: Equations (4.2.2) and (4.2.:1) follow directly from the backward version of Theorem 2.5.1 because t1b = b 0 y + 10 b and 1,(1) = 0, i,.(I) = id. We have

t1nb

= b @ y0(n-l) + 10 b 0 yO(n-2) + ... + 10 (n - 1) 0 b.

(4.2.4 )

Since h:(t;O",T,e)(I) = 0 only the first two summands of the right hand side of (4.2.4) survive, so that

k.t(b)(O", T, e) = 'Pt,-.(b)'Pt-t, (Y)I\(t 1 ; 0", T, e)(y) ... K(tn ; 0", T, e)(y) + 'Pt - t 1 (y )1\(t 1; 0", T, (J)( b)K( t-z; 0", T, (J)( y) ... K( t n ; 0", T, e )(y) which equals the right. hand side of (4.2.1). The structure of additive (-y, a)-white noise can now be described. Let V be a doublemodule *-vector space, i.e. V is an l-graded involutive vector space and also a CILmodule such that. d(A.v) = Ad(v)A- 1 and (A.V)* = v*A*. Then a homogeneous v EVe T(Vky,a) is (l,d(v))-primitive and, in combining Theorem 4.1.4 with Prop. 4.2.1, we have the following result.. 4.2.2. THEOREM. Let 7/; be a generator on the involutive (-y, a)-Hopf algebra T(V). We put (4.2.5) for v E V homogeneous, Then, in the vacuum state, the F t , t 2: 0, form an additive

(-y, a)-wlzite noise over V with generator 7/;. Let Lt(v) be the unique solution in doD of

(4.2.6) Then

Ft(v) = Lt(v)ft,=(ad(v))'(> The equations for a (a,,)-Hopf algebra T(V) are (4.2.7) and

Lt(v) - it(v) = Ft(v) - it(v) = it dA,(ad(v) - id)F,(v). We look at. the equation for d;lv. We have

87 and

=

it

+

+ dA

T(1J(v*))

Thus r t (d;;-l )L t (7J) equals the above integral and Ft (v) is equal to this integral by rOoo(O'd(v)) from the left. There is a remarkable consequence of the following

4.2.3.

(4.2.8)

+ JI1

ultiplied

Let B be a symmetric *-bialgebra and let b E B be such that b01 commutes with /lb. Then for any white noise (j.t) on B tile family is a family of commuting operators. PROOF:

PROPOSITION.

We have for s

:s; t j.(b)jt(b) = Lj.(b)j.(b(1))j.t(b(2)) = L

i, (bb(l))j.t(b(2))

= (M 0 (j. = (M 0 (j.

0 j.d)(b (1)(/lb) 0 i.d)(/lb)(b (1)

= Lj·(b(1))i.t(b(2))j.(b)

= jt(b)j.(b) where we only used the increment property, the condition [b 0 1, /lb] = 0, and the independence of increments. (> Applying this to T(V)(l',a), we see that Lt(v) form a commutative process for each Lt(v) which gives an other justification of why we prefer left white noise. We give a closer decription of the generators on T(V).

v

E V. In the next section we will see that this is not true in general for

4.2.4.

PROPOSITION.

Let D be a pre-Hilbert space and let

PO: V----. H(D) 7]0: V ----. D 1/10: V ----. C a: l ----. H(D) be such that Po and

1/10

are hermitian, a is a »..v) = a(>.)po(v)O'(>.)-l 770(>"V) = a(>')7]o(v) 1/1o(>'.v) = 1/1o(v)

(4.2.9) (4.2.10) (4.2.11)

88

[or all v E V aIld A E l. Then, hy recurrence, ih« equations

pW == Po; p(bc) = p(b)p(c) 11W

1/1(1)

(4.2.12)

= 'T/o; 1](bc) = p(b)l](C) + 1}(b)£(c) == OJ l,bW

= 1/)0;

1/1(bc)

(4.2.13)

= (T1W),T](c)) + 'ljl(b)£(c) + £(b)1/1(c) (4.2.14)

deiiuc a gencrator 1/1 on T(V) such that (p,1],a) PROOF:

rv

(pv"1]""a,,,).

Since T(V) is free, the procedure (4.2.12)-(4.2.14) defines a linear functional

1/1 on T(V). For b EKern t we have that 1/1(b*b) = 11r7(b)11 2 2:: 0 and 1/-' is conditionally positive. By assumption '!/Jo is hermitian, so that 1/1('0*) = 1/1(v). For monomials M of

length 2:: 2 we have 1/-'(M*) = 1jJ(M) by (4.2.14). It follows that 1/) is hermitian. The a-invariance of 1/-'0 implies 1jJ(A.'O) = 1/1('0). Furthermore, using (4.2.9) and the defining equation (4.2.12) for p, we obtain

p()...b)

= a()..)p(b)aPt 1

for all ).. E L, i

« T(V),

by induction on the length of monomials in T(V). Using this, (4.2.10) and (4.2.13) yield

1]()...b)

= ap)'T/(b) for all x E L, s « T(V),

and we have for b, c EKern t

1jJ(A.(bc)) = (1](b* .)..*), 1]()...c)) = (1](b*), a()..)-la()..)1](c))

= (1](b*), 1](c)) = 1jJ(bc)

which proves the a-invariance of 1/1. The mappping 1]",(b) l----+ 1](b) extends to a unitary operator U from the completion of D", to the completion of D such that

1]",

= 'T/

0U

p",(b) = U*p(b)U a",()..) = U*a()..)U.\> We have neglected one important aspect. Let 1jJ be a generator on the involutive B. In particular, B is a double-module *-vector space. The triplet (Pa,O' 1]a,O, 1/1a,O) with Pa,O = P", - tid, 1]a,O = 1]", and 1/1a,o = 'l/J is a triplet as it appears in Prop. 4.2.4 where a(A) is given by 'T/t/J (b) l----+ 1]", (a).. b). Hence there is an additive left white noise on the involutive vector space B given by the triplet (Pa,O, 1]a,O, 1/1a,O)' In other words, given a left white noise i.t on B we can associate with it an additive left white noise F t on B. Similarly, we associate an additive right white noise with a right white noise. Loooking at equations (4.1.5) and (4.2.5) and equations (4.1.9) and (4.2.7), we see that (4.1.5) can be written

h, a)-bialgebra

89

and (4.1.9) can be written

We also want simpler forward and backward equations for left and for right white noise respectively. Here we are facing the problem that left white noise is not forward adapted and right white noise is not backward adapted. Therefore, we multiply j,t(b) by r 0= (CYd(I.) )-1 from the left to obtain a forward adapted process This process satisfies

1.t.

].t(b)

= t(b)id + It (L (dA;( 7](

1 d(

b(l))b(2)))

+ d A,

)p( b(2))

- t( b(2) )id)

+ dA T(7](b(2))) + 1jJ(b(2))dr)) = t(b)id + It (L (dA;(7](CY d/b (2 d b(2)))

+

- tid)(b(2)))

+ dA T(7](b(2))) + 1jJ(b(2))dr)). If we multiply by rO=(CYd(b)) from the left we obtain

j.t(b) = t(b)id

+ +

- tid)(b(2)))

+ dA T(7](b(2))) + 'ljJ(b(2))dr)) where the integrals are to be understood in the obvious way. Comparing this with (4.2.8), we have

j.t(b) = t(b)id

+ It(jJT *dFT)(b)

which is in complete analogy to the symmetric case. In a similar way, we may write

for right whit.e noise. The relationship between j.t and Ft goes further. We will see that F t can be approximated by sums of the form 2:0-.lV)

= Q(v,w)

for all X E l,V,lV E 1/.

(4.3.1 )

Furthermore, let 'P be an a-invariant positive linear functional on T(1/). \Ve define the linear functional gQ,

j, (Xkl; Xlk = a((y*)-k )P>j,(XkL)

and (4.4.11) by We apply this to unitary q-white noise to obtain the result that all unitary q-white noise are of the form with U, a symmetric unitary white noise and with Aft the unitary process on en 12> F(L 2(1R+, H)) given by (Mt)kl = bklft(a)k.

We can build a multiplicative q-white noise out of a given symmetric multiplicative white noise just by multiplying by the M t- 1 - process provided the pre- Hilbert space is M-graded in such a way that L, I, Wand K are 'compatible' with this grading. In this sense, the class of multiplicative q-white noises, q =/:: 1, is smaller than the class of symmetric multiplicative white noises. For example, let Ut be a unitary q-white noise with H = C Then L, Wand K are all complex n x n-matrices. There is a /-lo E M such that H(p.o) = e and H(p.) = {o} for Il =/:: /-lo. If q is not a root of unity W and K must be of main diagonal form and L has zero entries except for the diagonal {(k, l) : k - l = /-lo}. If q is a 11t h root of unity Wand K have zero entries except for the diagonals {( k, l) : k - l == 0 mod 1I} and L has zero entries except for the diagonals {(k, l) : k - l == uo mod 1I}. A simple non-trivial example is obtained as follows. Let n be equal to 2 and H( -1) = Co Furthermore, put W = id, K = 0 and L = Then

103

with III the solution of

1 LL*dr)

2

is an example of a unitary q-white noise. We conclude with the remark that the method applied in this section also can be used to classify white noise on U(t) with l a finite group [Section 1.8, Example 6). 4.5. Cocomrnut afive white noise and infinitely divisible representations of groups and Lie algebras. The formulae for white noise become particularly simple for group-like and for Lie elements; see Prop. 4.1.2 and Prop. 4.2.1. In both cases, the comultiplication is cocommutative. We prove explicite expressions for white noise on Z2-graded cocornmutative *-bialgebras. These expressions do not involve quantum stochastic integrals or kernels but are convolut.ion products of the convolution exponential of a creation process, a generalized second quantization process, the convolution exponential of an annihilation process, and t.he I-parameter convolution semi-group of states associated with the underlying white noise. This generalizes the situation in the group *-hialgebra case where we can write

and the convolution product is just the ordinary composition of operators. As a consequence we have the result that the GNS-representation of the infinitely divisible state 'P = 'P1 embedded into the I-parameter convolution semi-group 'PI can be realized on Bose/Fermi Fock space. This generalizes the algebraic part of the so-called ArakiWoods embedding theorem for infinitely divisible positive definite functions on groups [8,9,32,63,85]; see also [71]. Moreover, it gives a classification of the infinitely divisible representations of Z2-graded Lie *-algebras; d. [56,86] where these representations are introduced. We begin with some general considerations. Let C be a douhle-module coalgebra such that C is a Cl-comodule and a Cl-module coalgebra and such that the compatibility condition d( A.e) = Ad( e)A-1 is satisfied. The symmetrization C'" of C is the coalgebra C 0 Cl with and

e..,(eA)

= e(c)

where we wrote eA for e 181 A. Suppose that we are given D, TI, p, '13 and .,p as in Section 2.3, and in addition a representation a : l ---+ H(D) of l on D such that

p(A.e) = a,\p(e)a"l TI( A.e) = a,\TI( c) 'I3(A.e) = a(,\*)-l'l3(e) .,p(A.e) = .,p(e)

(4.5.1 ) (4.5.2) (4.5.3)

(4.5.4)

104

t. We then can extend

for all c E C and /\ "

'I, p,O and -if; to

c"

by setting

1),(d)TI(C), p,.(d)

'VJ,.(d)

=0:

P(C)OA'

=

0'.\.19(c),

c_

-Ij;(c).

Next we apply Theorem 2.[>.1 to (II-" P_p 19'1J1P,.) and C,. and ohtain fa.milies

(: AD satisfying

= «c)icl +

jt (dA;(T7(C(2») + dA r(p(c(2»a A

-

t(c(2»id)

+ dAr (a_\d9(c(2)) + -Ij;(c(2»dr). Using Ito's formula and the conditions on '7, P,

(4.5.5)

and -,p, one can show that

also satisfy (4.5.5) and by the uniqueness of the solution of (4.5.5) in AD we have

Similarly,

-(a,,.)(( Cd\,-1) /\') = J.t

r s t (a A )J.t .(a,,.)( ce )

and, therefore,

=

(4.5.6)

Hence we have for j.t(c) =

(4.5.7) with and In the sequel we assume that C is a £'z-graded co commutative coalgebra. We also assume that D is a £'z-graded pre-Hilbert space and that Tt, p, and -,p are even mappings. If we put a(1) equal to the parity operator, i.e. a(l)fD(o) = id and a(l)fD(l) = -id, the tuple .(o,p,O,O)

J.t

an

d

satisfies conditions (4.5.1)-(4.5.4). We put j;t --.1 _

-(0,0,-.1,0)

J.t - J.t

.

=

=

105

equation

1.fi.1. PROPOSITION.

(4.5.8) holds for all (5, t) E T.

* 'Pt.

*

We show that

PROOF:

satisfies equation (4.5.7). By quantum Ito's

formula )() (). d .P ' 1 I J.t*J.t*J.t*'Pt-. C -eCl t = LU" (C(1»)f T( 00. For the same reasons the operators given by the kernels

= 1.;t

'(TI,P,","') 1.t

=

and

and have

B"t ) * 'Pt-. = (exp* BTI)'P st * 1.t * ( exp.••

where the convolution exponential series are to be understood in the sense of strong convergence on ED' We look for linear subspaces of ED which are left invariant by the jCll (c), c E C, and which contain the vacuum vector. The reason why we are interested in these subspaces lies in the fact that, in the case when the coalgebra is a &'2-graded *-bialgebra B, .,p is a generator on Band 17 = 17", 1 P = P"" {) = jj, we can embed the GNS-representation of the state 'P = exp".,p into the *-representation jOl of B by identifying the cyclic vector of the GNS-representation with the Fock vacuum. Remember that the elements of F(L 2(1R+, D)) can be regarded as sequences (Fn)nEI\l with r; E L 2(5n , D®n). The figure

5

T(D)

So

U

1

C

51

U

1

EI7

D

EI7

52

(D

U

1

(2) D)

EI7

shows a connection between F(L 2(1R+, D)) and the &'2-graded pre-Hilbert space T(D) consisting of all elements (2)nEI\lF(n) in the tensor Hilbert space EDnEI\lH0 n with F(n) E

ov-. In fact, T(D)

can be identified with the linear subspace of F(L 2(1R+, D)) formed by the functions w f--t F( w) that vanish outside of 5 n [0, 1] and are constant equal to some F(n) E D®n on 5 n n [0,1]' n E 1\1, via the mapping

which extends to a unitary mapping 'IT from the Hilbert space formed by all functions in F(L 2(1R+, H)) with support 5 n [0, 1) and which are constant on 5 n [0, 1] to the tensor Hilbert space over H.

108

Before we proceed with considerations on invariant. subspaces we introduce the 'graded' creation, preservation and an nih ilation processes. We form the graded tensor pre­Hilbert spilee

T(D)

=

EB D':)"

""oN

of D which is a linear subspace of T(D). The symmetric group S,. acts tensor product D'im of D with itself through

Oil

the n­fold

where are homogeneous elements of D of degree (/, and for (/ E: 71. 2 the factor sgn( 7f; (I, . . . ,(n) denotes the sign of the permutation of #{ l : (/ = I} elements derived from

by eleminating the numbers l in the upper and lower row for which (/ define the linear operator P D on T( D) by

=

O. We then

We define the linear operator

by

where (1, ... ,(k are the even and (k+I,' .. ,(m are the odd factors of the tensor •.• A'(O and R f-> A(R) are cvcn , linear preser-oaiion. operators. The mappings mappings from]) to H(S(D)) and from H(D) to H(S(D)) respectively. The latter also is hermitian. The mapping f-> .1(0 is an even, antilinear mapping from D to H(S(D)). For a decomposition D = D 1 EEl D 2 the canonical isomorphism between S(D) and S( Dt} 0 S( D 2 ) gives

+ id0.1*(6), .1(6)0id + id0.1(6), A(R 1 )0id + id0'\(R 2 )

EEl 6)

)0id

.1(6 EEl 6) A(R 1 EEl R2 ) for

E Dr,

6

E D2 ,

D 2 = D(1) we have

n, E H(D l)

and R2 E H(D 2 ) . In particular, for D 1 = D(O) and

A*(O

+ id 0 id + id 0

0 id

.1(0

0

where ED and and denote the even and the odd part of respectively. Notice that for an odd operator R in H(D) the operator A(R) does not leave invariant S(D(O») or S(D(l»). We have the following canonical commutation relations

[.1(0, A(()]

= [A*(O, .1* (()] = 0,

[.1(O,A*(()] = [A(R), A(S)] = A([R, S]), [A(R),A*(O] = A*(RO, [A(R), A(OJ = -A(R* 0 for ( E D and R, S E H(D) where the brackets are to be understood as graded commutators, i.e. [A, B] = AB - (_l)dAd B BA for A, B E L(S(D)). We equip the Hilbert space L 2(1R+) with the trivial graduation. For E D and R E H( D) we define operators on S( L 2 (IR+ ) 0 D) by

.1:t(O = .1*(X[.,t) 00,

= .1(X[.,t) 0 A.t(R) = A(X[.,t) 0 R) A.t(O

where, in the third definition, we identified X[.,t) E L 2 (1R+) with the corresponding multiplication operator on The operators A*(O, A(R) and A(O on the completion i(D) of S(D) are closable and the domains of their closures contain the linear subspace £(D)

= (

n

Dom c/L(id») n

U EE1'o(D)

T(E)

111

of J:(T»). 'I'he closures leave feD) invariant and we extend operators on

feD)'

We extend

Ji;t(O, A,t( R) and A,t(O

1*(0, A(R)

to operators on

and

A(O

to

f(L2(H 1)0D)'

1.5.4. PROPOSITION. The operators JJ01(c) and ng1(c) leave lJ1- 1(f(D») invariant and

B61(c)llJ1- 1(f(D») == lJ1- 1 0 A*(I)(C)) o lJ1, Bg1 (c)llJ1-1(f([)) = lJ1- 1 0 A(19(c)) 0 lJ1. PROOF: Using Lemma 4.1.1 we have for

that. (B61 (c)F(n»)(w) = 0 if #w

::p

n

-+ 1 and

for #w = n

-+ 1

(B61(c)F(n»)(w) =

L i

2) _l)d(c)(d(etl+---+d(e,

0 ... 0

0

T)(c)

0

0 ... 0

1=1

=

vn -+ lA*(I)(C))(L

=

(lJ1- 1 0 A*(1](c))

0

0 ... 0

lJ1)(F(n»).(> leave lJ1-1(f(D») mvarfan].

4.5.5. THEOREM. The operators

PROOF: From Prop. 4.5.4 and the fact that I:;:o h(B61)"I(c) strongly converges to j61(C) on f D we conclude that j61(C) leaves lJ1- 1 (.f(H )) invariant. Since j61(c) leaves £D invariant, we have that j61(C) leaves lJ1- 1(£(D») invariant. The same argument yields that jgl(C) leaves lJ1- 1(£(D») invariant. It remains to show that (p0n 0 F(n) is 1(£(D») 1(£(D»)' for F(n) E lJ1This follows from cocommutativity.O in lJ1We can realize on the Bose/Fermi Fock space J:(H). operat.ors on £(D) and (with the obvious notation)

The jOl(C) are

(4.5.11) We mention that, if 13 is a &'2-graded *-bialgebra and .p is a generator on 13, a white noise with generator .p can be realized on the Bose/Fermi Fock space .f(L 2(1R+, H)) and

i..

=

exp,,(A: t

0

* (exp*A,t oij)*cpt-o

2 with Pot ).[o,t) 0 P E H(L (1R + ) 0 D). The domain of jot(b) is This can be proved by checking the axioms (WNl)-(WN4) where cocommutativity must be applied and by using (4.5.11) to see that .p is the generator. A state cP on an involutive (-Y, a)-bialgebra 13 is called infinitely divisible if there are states CPn, n E 1\1, such that (CPn)"n = ip, A state of the form exp".p with .p a generator on 13 is infinitely divisible.

112

We state the following embedding tlieorcui taiine *-bialqc bT(J,$

JOT

inj£ni/cly diuisilil« slates on cocatnrtiu-

4.5.6. THEOREM. Let B be a lrgraded coconuuut.etive *-bialgebra. and let be a generator OIl B. 1'1/(>11 ilie GNS­representidioJI of .p =, exp.4" can be embedded into tlle »­represcutetiou j of B OIl [(1)) given by . exp,. ( 4' * J' =-

0

""v., ) * (,.C:)nEN P,j,(" n.

A) f\

0 Un

- ) *

jOl (b)f2

w11ic1l extends to a unitary mapping from H'I' to the closure of {jol(b)f2 : b E B} ill

F(H). Here (1r '1" H'I" f2'1') denotes ilie GNS­triplet associated witll .(rk)(p( adCb)) -

id)

k=l

if 17 = {r}, t" :S r otherwise. Thus

< t"+I, and

7

= {rl

< ... < rn} with r
00. Using an argument similar to that of the proof of Prop. 2.3.1, one shows that s"Ab) also satisfy the other conditions of Prop. 2.2.2., forms a classical sequence of random variables. These random variables will not be ordinary sums of independent identically distributed random variables but the deformed increments X mn = X n - Xmb mn (where we assumed q E IR for

137

simplicit.y!) arc independent in the sense of non-commutative probability theory. In the state 1>, they give rise to a seq uence of independent classical random variables. However, the families (X m n ) and (X m ) do not commute in general, and it. will not always be possible to realize them simultaneously, in the state 1>, as classical random variables. Our discrete processes also sat.isfy certain quantum stochastic difference equations. For instance,

The basic example is when Do fl

=

1. Then D

=

C2 , a

=

C, D 1

=

b

=

=

C, D"

= {O}

for n rf- {O, I},

at Do = id

and

In this case, t.he above difference

equation becomes

which can be regarded as the discrete version of the quantum stochastic differential equation for quantum Azema noise. The classical sequence (X n ) is the classical Markov chain that led P.A. Meyer and K.R. Parthasarathy to the quantum stochastic differential equation of the Azerna martingale; d. [58,61]. 1 1 • Theorem 6.3.2 t.ells us that the normalized sums /7;; converge to

vn

2-dimensional Azema noise (L t, L;) and to the interpolation (L t, L;) of 2-dimensional quantum Brownian motion respectively with the covariance matrix given by

Q=

(lIa*oflW

0)

Ila[211 2

.

Moreover, converges to (Ltft,OCl(b q ) , L;ft,OCl(bq ) ) by Theorem 6.3.1. We conclude with the treatment of the case q = O. Assume that we are given a quantum probability space (A o,