Proceedings of the Conference Quantum Probability and Infinite Dimensional Analysis : Burg (Spreewald), Germany, 15-20 March, 2001

where Ml = Mfc(C), a fixed full matrix algebra. Suppose we have a locally faithful state y? on 21 with restrictions y>A, A C Zd bounded, to local algebras 2U. Consider a pair A CC A of bounded regions of Zd. Denote %\A

the (/^-conditional expectation relative to the inclusion 21A> A C 21A. Definition 2.1. The locally faithful state if € <S(2l) is said to be a Markov state if for every pair of regions A, A as above, we have (2)

Condition (2) simply means that -£/-?. acts trivially on the observables localized in A\A. It is of interest to formulate the quantum Markov property in terms of a net of Umegaki conditional expectations which leave the state

ETA\A V(E?V* ;) f c . X . :=lim„ Z^v A\A n

k=Q

(3) '

V

5 By Theorem 5.1 of 2 , such conditional expectations project onto the largest expected subalgebras of 2l^\A> and leave invariant by constuction the state (p. Moreover, by the Markov property (2), the ^\A factorize as

where £

A

: 2lx t-> Q3^A C 2laA is given by

In order to understand the infinite-volume theory, one should investigate the behavior of Q3gA = 7^(£A\ A ) as increases in order to exhaust oil ofZ d . Theorem 2.2. (Theorem 3.3 of '4) Let if G <S(2l) be a Markov state. For each bounded region A C 1,d there exists a Umegaki conditional expectation E^ on 21, which projects into 21A' . Moreover, the net {£^A'}Aczd satisfies: (ii) EK [ax/ = idsax, , (Hi) (p o EK = (f, (iv) if AI C A2 then

Proof. We report only a sketch of the proof given in 4. By Theorem 5.1 of 2, it follows that the ranges 58 gA of the £A> A give rise to a decreasing net of subalgebras of 2laA; taking into account (4). Namely, {^M A} ADD A ^s a decreasing net of conditional expectations defined on the full matrix algebra 21A, which converges to a conditional expectation by a standard martingale convergence theorem, see 22 , Theorem 3. Denoting

the expectation £A< projects ontoa ==

a

f)

The Umegaki conditional expectations {£A/} are called sometimes transition expectations.

6

and gives rise to the searched conditional expectation

The net {£'A'}ACZ<' is projective, and leaves by construction the state tp invariant. D In order to investigate the converse of Theorem 2.2, we consider the potential associated to a locally faithful state

(5)

The potential {/IA}ACZ<' satisfies standard normalization conditions, together with compatibility conditions. In addition, if the state

(6)

for selfadjoint operators H^ A e 2l^> A , K-^ € 21A, and M^A e 2tgA- Further, the above operators satisfy the following commutation relations 0,

(7)

see 4, Lemma 4.1. Now we are in position to give the converse of Theorem 2.2. Theorem 2.3. (Theorem 4.2 of*) Let be given a locally faithful state

7 where fc^ A is the transition operator given by fc XA = e~5' l A e i' 1 A\A Thus, in our situation we get

that is E~* acts as the identity on 21?A\\ A. . A\A

D

3. The relation with statistical mechanics We start with a Markov state yj on the quasi-local algebra 21. The potential {/IA}ACZ<J associated to

(8)

exists and defines a strongly continuous one-parameter automorphisms group of 21. By constuction, tp is automatically a KMS state for at (i.e. a KMS state at inverse temperature (3 = — 1). In addition, tp is faithful, see 4 , Section 5. The disintegration theory of states which are invariant w.r.t. a net of conditional expectations which act locally, was developed in 1. For such a disintegration, one should start by considering the natural extensions eA, of the E£ to all of ^(21)" given by

The projections {eA,}Aczd are then the cyclic projections relative to the ranges 7^(e^,) of the e1^,. The orthogonal measure (19, Section 3) corresponding to such a disintegration is precisely the {^(21), {e^,}} '-measure, see 1, Theorem 3.2. In addition, a Markov state tp is also a KMS state, then one can use standard results relative to the disintegration of a KMS state into states describing pure phases, see e.g. 9. The main result is that both disintegrations coincide. Namely, in the situation under consideration we have

where, for A bounded, 3£ := /\

8 is the algebra at infinity, and 3V :=7r ¥ ,(a)'A7r v ,(2l)" is the centre of the GNS representation of the state (p. We collect the main properties of a Markov state in the following Theorem 3.1. (Theorem 5.4 o f 4 ) Let (f 6 >S(2l) be a Markov state. (i) The state tp is a {at}-KMS state, where at is the automorphisms group given in (8), (ii) the state (p admits a disintegration
1>n(d1>)

(9)

JS(<& S(<&)

where the measure n is concentrated on the factor states of 21, (Hi) Almost all ty appearing in (9) are {at}-KMS states, and Markov states in the sense of Definition 2.1. To conclude, we shortly describe the possible appearence of the wellknown phenomenon of symmetry breaking. In order to do this, we consider translationally invariant Markov states. Let rx : 21 i—» 21 be the shift generated in a natural way on the spin algebra by the translation x £ Zd, and consider a Markov state tp e <S(2l) invariant w.r.t. {rx}x&Zd. One easily verifies that the generalized conditional expectations associated to the restrictions of

*'

which yields for the corresponding Umegaki conditional expectations given in Theorem 2.2, EA, = r- 1 o£ A ,+x°r x . d

(10)

n

Consider the dual action {Tx}x&i ° the state space of 21. It is straightforward to verify that Tx maps the set SE C <S(2l) of all {£A'}~invariant states into itself, leaving globally stable the set dSe of its extremal states. We easily conclude by the uniqueness of the maximal measure ( 7 , Theorem II. 3. 6), that Tx preserves the measure n given in (9): H°Tx=v

(11) d

We then have a measure-preserving Z -action x H-> Tx on the state space 5(21) of 21.

9 Fix our attention on an ergodic component m of the ergodic disintegration = / mv(drri) of the measure fj,.b If the standard measure space (5(21), m) is essentially transitive for the action of Zd (that is when m is concentrated on a single orbit), then the Zd-space (5(21), m) is similar to the Zd-space (Zd/H,\) based on the homogeneous space Zd/H, see 17, Theorem 4.12 (see 16 for the original result). Here H C Zd is a subgroup of Zd, and A is a probability measure equivalent to the Haar measure on the Abelian group Zd/H. In the {Eh>}-extremal disintegration of a {r^J-invariant Markov state 9'18 for quite similar situations and for technical details. 4. Markov states on non homogeneous chains: general properties General properties of quantum Markov states were firstly investigated for one-dimensional models where the order plays a crucial role. Recently, taking into account results contained in 5|6, the emerging structure has been fully understood, see 3. Here we report the main results relative to quantum Markov states on chains. We start by considering a totally ordered countable discrete set / containing, possibly a smallest element j_ and/or a greatest element j+. Namely, if / contains neither j_, nor j+, then / ~ Z. If just j+ e /, then / ~ Z_, whereas if only j_ S /, then / ~ Z+. Finally, if both j_ and j+ belong to 7, then I is a finite set and the analysis becomes easier. If / is order-isomorphic to Z, Z_ or Z+, we put symbolically j- and/or j+ equal to —oo and/or +00 respectively. In such a way, the objects with indices j_ and j+ will be missing in the computations. The bounded sets Afcj := [k, j] assume a fundamental role in the sequel. In this situation, the spin algebra is the quasi-local algebra 21 obtained by the infinite C*-tensor product of full matrix algebras {MJ}je/ with The ergodic disintegration of the measure fj, corresponds to the direct-integral disintegration ( 21 , Theorem 8.21) of the von Neumann algebra L°°(<S(2l), fj.) w.r.t. the fixed-point subalgebra relative to the natural Zd-action on L°°(5(2l),/j).

10

possibly different dimensions: Mj :=M^.(C). Notice that the present situation naturally arises from the previous one when we restrict ourselves to any increasing sequence of bounded regions {r>c}fceN of 1>d satisfying

and exhausting all of Z^:

ljr fc = z d .

(12)

fceN We have,

and / ~ Z_ .° We start with a locally faithful state

the if\k , +1 -conditional expectation relative to the inclusion 2lAfc , C Definition 4.1. The locally faithful state y> 6 5(21) is said to be a Markov state on the chain if for every k, I e I with k < I, we have

Also in this situation, we can find a net of Umegaki conditional expectations which act locally, and leave fixed the Markov state (p. This can be done by recovering a very explicit structure of the potential associated to VTheorem 4.2. (Theorem 5.1 o f 3 ) Let (p 6 5(21) be locally faithful. Then the following assertions are equivalent. (i) (f is a Markov state on the chain. c

According to the previous literature ( 5 ' 6 ), we are using the reverse order.

11 (ii) For each k < I, the potential { h / ^ k l } associated to if by (5), can be recovered by i-i h^^Hk + ^Hjj+i + Hi, (13) j=k from sequences {#j}j_<j<j + , {tfj}j_<j<j+ and of selfadjoint operators localized in 21^.., j and 21^ j+1 respectively. Such sequences satisfy the commutation relations [Hj,Hj]

= 0, [Hjj+i,Hj+i] — 0, [Hjtj+i,Hj+ij+2]

= 0, = 0.

(Hi) For every k < I < j there exists a sequence of Umegaki conditional expectations {-Efc.j} defined on the algebras {2tAt J+1 }- Such a sequence satisfies

Ekj [aAliJ. = Ei,j i and leaves invariant the state ip: <^AfcJ ° Ek,j = Mj+l H^ K(£j) c Mj . Namely, E^ida^.^®^ where

(14)

12

It is straightforward to show that such transition expectations satisfyd £j-l(A®B) = £j-l(A®£i(B®I)),

j-<j<j+.

(15)

Finally, we remark that also the last situation can be fitted into quantum Markov processes on directed sets. Namely, the directed set is precisely {Afc,j}j_
(16) and define 58:= (®j,<j<j+Bj) ® Mj+ . Then we obtain in a canonical way, a conditional expectation

d

ln the most general case of non locally faithful states considered in the next section, relations (15) could be not satisfied in general.

13

defined to be the (infinite) tensor product of the following conditional expectations a e Mi

(17) ^ '

>-* V Pi^jaPi'*'J / y

together with the identity map on MJ'+. The reduced algebra

Ml, =R can be written as

MPJ = #4® Aft,j j ~uj

(18)

with A^. and N£. all finite-dimensional factors. Again, the states <^ on N-ij. ® AP+1 are uniquely recovered by the transition expectation
(19) whose law fj, is uniquely determined by the initial distribution and transition probabilities given respectively by <_ :=^ A ,_,,_(Pij_))

(20)

The measure space (Q, /u) is obtained as the projective limit of compatible measure spaces {(^A, (J-A.)}A.C!J we denote by ^A : fi i-> fiA the canonical projection of fi onto QAJ Let OQ C fi be the set consisting of those w 6 0 such that all 7rlAjj ,(V), together with TT] A , , A , ., are nonvanishing. the set QQ is a mea" j,j ^ ''" j+i,j'+i *• •* surable set of full ^/-measure. Consider, for each w € fi, the (infinite) tensor product Q3W given by _)®M^+

(21)

A completely positive identity-preserving map E^ : 21 >—> iBw is uniquely defined as the (infinite) tensor product of the maps « e M ^ .w

,

(22)

14

together with the identity map on MJ+ . We have trivially EwoE = Eu

(23)

where E is obtained by the (infinite) tensor product of the maps given in (17). Denoting (with an abuse of notation) by ujj the canonical projection u on !BW given by

determined by the initial distribution, which is the state on N^_ given by

);

....

(25)

by the states rfu.^.+1 on A^,,. N^^ , given by

and by the final distribution which is the state on N^ _t <8> M-7'4- given by ^-\ ^ ® B) :=
(27)

Finally, we recover a sequence {£u>:'}j_<j<j+ of conditional expectations

given by 3 £,"•*J (\(\ a ® CL]/ <& (b ® b)) = nli*Jj ^^ ^ \ //

,{jjj-^l

(a K> b)n^^ (b ® Da, &> / ' \ / Hjjj^. i ,Wj-J-2 *• '

It is easy to verify that the states i/ju, u> £ QQ, are Markov states on 03^, w.r.t {fo;J'}j_<j<j+ given by (28), respectively. Further, the map w e fi0 ^ Vu; ° Eu e 5(21)

is cr(2l*,2l)-measurable. We are ready to report the announced result concerning the disintegration of a Markov state into elementary Markov states which are minimal

15 in the sense that the ranges of the associated transition expectations have a trivial centre. Theorem 5.2. (Theorem 3.2 o f 3 ) Let
ipuH(duj} (29) Jsi where u> £ 17 H-> tpu €E <5(2l) is a cr(2l*,2l)-measwra6/e map satisfying, for all u 6 n,

The proof of the above theorem relies on standard techniques of the theory of direct-integral decomposition of representations of C*-algebras. Such a proof can be found in 3. Notice that the GNS representations 7rVij of (pu give rise to von Neumann factors. The proof of the last assertion follows from 8 , Theorem 2.6.10. Further, the disintegration (29), even if it is made of factor states, does not correspond to the central disintegration given in Theorem 3.1. The following theorem is the converse of Theorem 5.2 and can be also regarded as a reconstruction result for quantum Markov states on chains. Consider for j_ < j < j+ , a sequence Z? of commutative subalgebras of J M with spectra fij and generators {-F^.j^en.,; a Markov process on the product space

n== with law n determined, for ujj 6 Qj, ujj+i e fij+i, by all marginal distributions Tri , and all transition probabilities 7ri. >w . For (jjj e fij such that TT£ > 0, fix a splitting as (18) M£, = A £ . ® t f 4

^j

by finite-dimensional factors. For Wj_ e flj_ such that 7ri~_ > 0, choose a initial distribution 7yi^_ on •jN

16

For each pair (ujj,ujj+i)

£ 0, x n,-+1 such that iri,jtU.+l > 0, consider a

For Wj + _i € fij+-i such that ^ _^ > 0, consider a final distribution Then, on the measurable set Q.Q of full /^-measure consisting of sequences (jj such that all the n3 , ,, and Trj , . , . are nonvanishing, the state V>w in (24) is a well-defined Markov state on the quasi-local algebra QSu, given in (21) w.r.t. the sequence {£L/}j_<j<j+ of transition expectations (28). Finally, defining £L, : 21 1-> »„ by (22), the map ^A7.1*-

" A 7 . 1 \ W J ' 9 A ^ i 1 ; i 1 \W)

•*

w 6 QO >-> V»u ° -EL, =: yw € «S(2l) ,

(30)

is cr(2l*,2l)-measurable.

Theorem 5.3. (Theorem 4.! o f 3 ) Let ipu be the measurable field on 5(21) given by (30). Then the state

:= / Jsi0

is a Markov state w.r.t. any sequence {£ J }j_<j<j + of transition expectations with Z(7Z(£i)) = Z?, determined according to (1), by states
(5

2 _ (6 <

The proof of the last theorem consists in showing that the state ip is a Markov state w.r.t. any sequence of transition expectations {£J} constucted by (1), taking into account (31). The reader is referred to 3 for the complete proof. 6. Quantum Markov states on general quasi—local algebras The investigation of the quantum Markov property for multi-dimensional spin systems suggests us the natural definition for Markov states on general

17 quasi-local algebras. A quasi-local algebra (8, Definition 2.6.3) is a C*-algebra 21 obtained by the C*-inductive limit of a net {2la}ae>i of C*-subalgebras with the same identity. The directed set A has also an orthogonality relation J_ such that {<)},

a 1/3.

(32)

For each a G A, one can define

Oa' := V a/» j8J_a

where the bar denotes the uniform closure. In the previous situations, A = {A C Zd, A bounded } and AI J. A2 if AI n A2 = 0. This general situation covers also cases arising from Quantum Field Theory where the a consist of bounded regions of physical space-time, and the orthogonality relation _L describes Einstein causality, see e.g. 13. Further, the commutation relations (32) can be suitably replaced in order to include Fermion algebras, or field algebras with different commutation relations. We are ready to give the definition of the quantum Markov property for the general situation of quasi-local algebras.6 Let 21 be a quasi-local algebra together with the local filtration {21Q}Q6,4 as above. Suppose that for each a G A, another index a £ A is assigned with a -< a. Definition 6.1. A state ip 6 <S(2l) is said to be a Markov state if there exists a filtration f {25Q'}aeA of C7*-subalgebras of 21 such that

2lg- C *BQ' C 21Q<, together with a projective net {Ea>}a€A of Umegaki conditional expectations such that (i) EC, : S l ^ < B a ' ,

(ii) }a£A by more manageable properties, we need additional e f

We are indebted to V. Liebscher for fruitful suggestions about this point. This means that a -< /3 => *Bpi C 25a'.

18

conditions on the quasi-local algebra 21, as well as on the state

19

19. Sakai S. C*-algebras and W*-algebras, Springer, Berlin-Heidelberg-New York 1971. 20. Stratila S. Modular theory in operator algebras, Abacus Press, Tunbridge Wells, Kent 1981. 21. Takesaki M. Theory of operator algebras I, Springer, Berlin-Heidelberg-New York 1979. 22. Tsukada M. Strong convergence of martingales in von Neumann algebras, Proc. Amer. Math. Soc. 88 (1983), 537-540.

QUANTUM BOLTZMANN STATISTICS IN INTERACTING SYSTEMS

LUIGI AGCARDI, SERGEI KOZYREV Centra Vito Volterra, Universita di Roma Tor Vergata Abstract. Collective operators for generic quantum systems with discrete spectrum are investigated. These operators, considered as operators in the entangled Fock space (space generated by action of collective creations on the vacuum) satisfy a particular kind of quantum Boltzmann (or free) commutational relations.

1. Introduction In the present paper we investigate the statistics of the interacting (or entangled) operators in the stochastic limit of quantum theory1. We investigate the model of a quantum system interacting with the reservoir (quantum field). The corresponding Hamiltonian is a combination of interacting, or entangled operators. An interacting operator is a product of an operator of the system and an operator of the reservoir, for example the product of annihilation of the reservoir (bosonic quantum field) and a system operator D*a(k). In the stochastic limit the quantum field becomes a quantum white noise. We will show that for the stochastic limit of discrete quantum systems interacting with quantum fields (for details see 2 ) the statistics of entangled (or interacting) operators is the particular variant of quantum Boltzmann, or free statistics. Analogous behaviour (arising from free statistics) was already found for particular system Hamiltonians with continuous spectrum3'4'5'6. In the present paper we will investigate generic discrete quantum systems interacting with quantum fields. Definition. A quantum system with Hamiltonian HS acting in the Hilbert space HS will be called generic, if: 21

22

i) The spectrum Spec HS of the Hamiltonian is non-degenerate. ii) For any positive Bohr frequency w > 0 there exists a unique pair of energy levels £iw,£2UJ G Spec HS such that

0.

The term generic means that the eigenvalues of HS are irregularly displaced. For example, the spectrum of the 1-dimensional harmonic oscillator satisfies (i) but not (ii). Thus it is not generic in the above sense. In the stochastic limit, cf. l , every positive Bohr frequency (a difference of eigenvalues of a system Hamiltonian) gives rise to a quantum white noise. In the work2 the authors showed that, in the stochastic limit, the dynamics of a (generic) quantum system is described by a stochastic Schrodinger equation which contains the following combination of quantum noises and system observables (that we call interacting, or entangled, or collective operator): <*„(«, fc):=|2w){l«|®&u,(*.*0(1) where the quantum white noise, cf. *, satisfies the following (bosonic) relation [bu(t, Ar), &„,(*', k')} = 2mW<5(* - t')S(w(k) - w)6(k - k').

(2)

Actually, even before the limit the evolution equation in the interaction picture contains terms of similar kind

This suggests that Skeide's analysis of the stochastic limit in terms of Hilbert modules7 can be extended to the present case. In the present paper we show that the operators (1) satisfy a variant of the quantum Boltzmann (also called free, or infinite) commutation relations: did,* = 5ij.

(3)

The quantum Boltzmann (or free] algebra is generated by the a.,-, a£, called creation and annihilation operators, with the relations (3). No other relations are assumed and different creators a*, a^ do not commute. Therefore, vectors in the Fock space of the type

23

are distinguishable and that justifies the name quantum Boltzmann relations cf. for example10'11'12. Generalizations of such relations were found in the large N limit of quantum chromodynamics6, in models of particles interacting with a quantum field (which include quantum electrodynamics and the polaron model)3'4.

2. Quantum Boltzmann statistics for entangled operators Given a generic quantum system, and a Bohr frequency u>, denote by \l^) and |2W) the two eigenstates corresponding to the two energy levels, eia, e^, so that

Hs\l*)=£iM

, HS\2U) = e2a\2u)

, w -£2u - £la > 0. (4)

With these notations we see that the restriction of HS to the space generated by |l w ) and \2J) is £2 0

0 £l

We define for each positive Bohr frequency w the Pauli matrix that flips the spin down by (5)

Using this we define the interacting (or entangled, or collective) operator

by c^(t,k) :=\2u)(lul\®bu(t,k)=a+®bu(t,k).

(6)

Remark. The operator (6) defined for non-positive Bohr frequency ui < 0 is equal to zero, since the quantum white noise bu(t, fc) for w < 0 is equal to zero. One can get the following relation on the collective operator, cf. 9 cl(t,k)=0

(7)

In the present paper we will investigate the case of many Bohr frequencies and get for collective operators the quantum Boltzmann relations. Remark. In the paper9 operators of the form (6) were considered for the case of a single Bohr frequency and their vacuum statistics was shown to satisfy Boolean independence in the sense of von Waldenfels. The present

24

paper extends this result to the case of multiple Bohr frequencies and establishes a connection between Boolean and Boltzmannian (or free) independence. It is interesting to notice that these types of independences arise naturally, i.e. from physically meaningful objects, in a purely bosonic context. The dynamics of a system in the stochastic limit is described by the white noise (or master) Hamiltonian which in the considered case takes the form

h.c.

I '

(8)

1

In other words: if HS is generic, any generalized dipole interaction Hamiltonian of 5 with a boson field is equivalent, in the stochastic limit, to a (possibly infinite) sum of independent 2-level-systems. The summation in (8) runs over the set of all Bohr frequencies. The simplest case corresponds to a single 2-level-system (one Bohr frequency), or the spin-boson model, cf. 8. Notice that the operators a^, b^ enter into the master Hamiltonian only through the combinations

(9)

and therefore the basic dynamical quantities like the propagator Ut, the wave operators 17±, the scattering operator S, will depend only on these combinations. This suggests to consider the algebra generated by the entangled operators cu(t, k) := a+ ® bu(t, k) ;

c*u(t, k) := a~ ® 6* (t, k)

Then all the calculations involving only matrix elements of the iterated series expansion of Ut, the solution of the white noise Hamiltonian equation, cf. l

dtut = with initial condition UQ = 1, can be done entirely within this algebra. A representation of the entangled algebra can be obtained within the Fock module F which is the submodule of

25

where .Fmast is the Fock space of the master field, B(7is) is the algebra of bounded operators in the system Hilbert space Us- The Fock module .^ent is a linear span of the entangled number vectors: II £„(*»' MO) n

where |0) is the Fock vacuum of the master field. Theorem. The operators cu(t, k), c^(t, k) considered as operators on the Fock module Fent of entangled number vectors satisfy the relations module Cu(t,k)c*u,(t',k')=2Tr8w5(t-t')6(k-k')6(w(k)-u;)(T+o--.

(10)

The operator
Using the commutation relation (2) we get

^(t,fc)c:,(^fcO = 27rf Wfc ,,J(t-0*(*-AO*M^ Let us show that the operator

is equal to zero on the Fock module ^"ent. To check this we consider the action of (11) on a number vector of (12) n=l

The action of (11) on such a vector will be non-zero only if in the product (12) at least one of the creators is equal to <£,(£, k). Let us denote by IQ the index of the first creator, starting from the left with this property. We get ff ff

£ u'

II °Zn°Z I
II CT^n combination of b* n (t n ,fc n )|0). I0
But for a generic system the combination

2

<7>j< n ^j = i ->(i<>j' n
equals to zero since (!<>:' I]

ff

««IU=0.

(13)

26

To prove this let us note that for generic Hamiltonians every operator a~k acting on the eigenvector \lu) of HS (with the eigenvalue e) kills it or maps into the eigenvector \2U) with the eigenvalue £ — w. Since arbitrary operator a~k decreases the energy (this is exactly the place where the stochastic limit procedure is important, since in the stochastic limit only the terms with positive Bohr frequences in interaction Hamiltonian survive), and the product of cr~k in (13) is non-empty, the result of application of the product of a~k in (13) to (l^) can be either zero or a vector orthogonal to |lu,). Therefore, the matrix element (13) is equal to zero. We checked that the second term (11) vanishes what finishes the proof of relations (10). Remark. One can find an alternative representation for the relations (10), defined by the operators (14)

where c(t), c^(k) satisfy the relations c(t}c*(t'}=5(t-t')

- k').

(15)

This gives another representation of (10) using the tensor product of quantum Boltzmann algebras. An analogous representation for the QED (quantum electrodynamics) entangled algebra was proposed in 4. Moreover, the operator c w (fc) in (15) can be formally reproduced by the formula where c^ and c(fc) are Boltzmannian annihilators. Remark. In the paper5 the operator valued quantum Boltzmann relations arising in the stochastic limit were called entangled, or interacting commutation relations. Remark. In 5 a second quantized version of the module quantum Boltzmann relations have been shown to be universal in any interaction with conservation of momentum. The results of the present paper show that such relations arise in every generic open quantum system. This rises the problem of extending the above construction to non-generic systems thus obtaining a classification of all the possible commutation relations arising in the stochastic limit of quantum mechanics.

27

Remark. The term entanglement is usually referred to states and expresses the impossibility of representing a state vector of an open system as a tensor product, i.e. in the form r system 09 ^r reservoir'

Such states are unstable under the evolution of interacting systems which creates linear combinations of them, i.e. states of the form ^system ™ ^reservoir

n

(with a number of terms strictly greater than 1) these are the entangled states. In the stochastic limit, the evolution of generic interacting systems (that creates entanglement) is described by quantum Boltzmann relations of the type (10) for collective operators (system plus reservoir). Therefore we might say that entangled states present the Schrodinger picture of entanglement while the entangled relations correspond to the Heisenberg picture of entanglement. Acknowledgements The authors are grateful to I.V. Volovich and M. Skeide for discussions. Sergei Kozyrev is grateful to Centro Vito Volterra and Luigi Accardi for kind hospitality. This work was partially supported by INTAS 9900545 grant. Sergei Kozyrev was partially supported by RFFI 990100866 grant. References 1. Accardi, L., Lu, Y.G., Volovich, I.V., Quantum Theory and its Stochastic Limit, Springer Verlag, "Texts and Monographs in Physics" (2001) 2. L.Accardi, S.V.Kozyrev, Lectures on quantum interacting particle systems. In: QP - PQ Quantum Probability and White Noise Analysis, Vol XIV, 1-196, World Scientific, 2002, Preprint of Centro Vito Volterra, 2000, Rome, partially published in quant-ph/0005029 3. Accardi, L., Lu, Y.G., Volovich, I.V., 'Interacting Fock spaces and Hilbert module extensions of the Heisenberg commutation relations. Publications of HAS, Kyoto, 1997. 4. L. Accardi, S.V. Kozyrev, I.V. Volovich, Dynamical g-deformation in quantum theory and the stochastic limit, J. Phys. A 32 (1999), 3485-3495, qalg/9807137 5. L. Accardi, I.Ya. Aref'eva, I.V. Volovich, N on-Equilibrium Quantum Field and Entangled Commutation Relations. Special Issue of Proc. of Steklov Mathematical Institute dedicated to 90th anniversary of N.N.Bogoliubov 6. I.Ya. Aref'eva, I.V. Volovich, Nucl. Phys. B 462 (1996), 600-615

28

7. M. Skeide, Hilbert modules in quantum electro dynamics and quantum probability. Commun. Math. Phys. 192 (1998), 569-604 8. Accardi, L., Kozyrev, S.V., Volovich, I.V., Dynamics of dissipative two-level system in the stochastic approximation. Phys. Rev. A 57, N3 (1997), quantph/9706021. 9. M. Skeide, A central limit theorem for Bose Z-independent quantum random variables. Infinite Dimensional Analysis, Quantum probability and related topics 2 No. 2 (1999), 289-300 10. J. Cuntz, Commun. Math. Phys. 57 (1977), 173 11. R.Speicher, Lett. Math. Phys. 27 (1993), 97 12. D. Voiculescu, D. Dykema, A. Nica, Free Random Variables. CRM Monograph Series, Vol.1, American Math. Society, 1992

STATIONARY QUANTUM STOCHASTIC PROCESSES FROM THE COHOMOLOGICAL POINT OF VIEW

GRIGORI G. AMOSOV* Department of Higher Mathematics, Moscow Institute of Physics and Technology, Dolgoprudni 141700, RUSSIA E-mail: [email protected]

The stationary quantum stochastic process j is introduced as a *-homomorphism embedding an involutive graded algebra = $El Ki into a ring of (abelian) cohomologies of the one-parameter group a consisting of *-automorphisms of a certain operator algebra in a Hilbert space such that every x from Ki is translated into an additive i - a-cocycle j(x). It is shown that (noncommutative) multiplicative Markovian cocycle defines a perturbation of the stationary quantum stochastic process in the sense of such definition. The Eo-semigroup on the von Neumann algebra N associated with the Markovian perturbation of K-flow j posseses the restriction PIN^, No C N, which is conjugate to the flow of Powers shifts P associated with j. It yields for P an analogue of the Wold decomposition for classical stochastic processes on completely nondeterministic and deterministic parts. The examples of quantum stationary stochastic processes on the algebras of canonical commutation, anticommutation and square of white noise relations are considered. In the model situation of the space L2(W) all Markovian cocycles of the group of shifts are described up to unitary equivalence of perturbations.

p

1. Introduction.

Let v be a measure on the real axis R. Let S = (St)tEwdenote the flow of shifts acting on the measurable functions f on W by the formula (Stf)(x) = f (x - t), x, t E W. Then, given a number r E R the function Ir(t) = v([r,r+t]), tEW,isal-S-cocycle,i.e. I,(t+s) =Ir(t)+St(Ir(s)), t , s E R. Analogously, fixing numbers r l , . .. ,rr, E W we get that the function Irl ...rk(tl, . . . ,tk) = I T l (tl)Stl (Ir2(tz)St2(. . . ITk (tk) . . .), ti E R, associated with the tensor product vBk is a k - S-cocycle with the characteristic property Stl (I(tz,. . . ,tk+l))- I(tl+tz,t3,. . . ,tk)+. . . + ( - l ) i I ( t ~ , . .. ,ti-1, ti+ ti+ll ti+zl.. . ,&+I) . .+(-l)kI(tll.. . ,tk) = 0, ti E R. One can consider

+.

*THE WORK WAS SUPPORTED BY INTAS-00-738

29

a ring of cohomologies H * = @Higenerated by the measure v in this way. The canonical bilinear map (xi,xj ) -+ xi U xj E Hi+j , xi E H i , xj E Hj , defining a ring structure in H* can be obtained from the action of S by the formula (xi U x j ) ( t l , . . . ,ti+j) = xi(tl1.. . ,ti)Stl+...+ti(xj(ti+ll.. . ,ti+j)) (see 14). We consider a graded algebra of cohomologies A = $ z l A i such that Ai consists of additive i-cocycles of the group of automorphisms a associated with stationary quantum stochastic process j over an involutive algebra K . In our construction A1 is generated by the basic operator-valued stochastic measures which are the creation, anihilation and number of particles processes in the important applications. Given an involutive algebra K we put K1 = K and construct the graded algebra K = @ z l K i with respect to a certain linear associative operation @ : Ki x Kj -+ Ki+j. In the case when K is a Lie algebra one can choose the universal enveloping algebra for K. We define a stationary quantum stochastic process as a *homomorphism j of K into the standard ring of (abelian) ~~homologies of the group a such that every x E Ki is mapped to an i - a-cocycle j(x). The homomorphism j transferes the operation @ in K to the cohornological multiplication U. We consider a class of cocycle perturbations of j by Markovian cocycles. The Markovian cocycle perturbation of K-flow constructed through our procedure determines the associated Eo-semigroup on the No c N, von Neumann algebra N which posseses the restriction conjugate to the flow of Powers shifts associated with the initial K-flow. Using the technics of lo, it is possible to extract an automorphic part of the Eo-semigroup. Hence our result defines an analogue of the Wold decomposition for classical stochastic processes on completely nondeterministic and deterministic parts. We give several examples where j determines quantum stochastic processes on the algebra of canonical commutation or anticommutation relations and the square of white noise correspondingly. The basic ideas are illustrated on the model of Markovian perturbations for the group of shifts in L2(R). In this case the associated Markovian cocycles are constructed in the explicit form by means of the inner function techniques.

p(No,

p

2. Markovian perturbations of the group of shifts in L2(R). Let S = (St)tEwbe a strong continuous group of unitary operators on a Hilbert space 'FI. A strong continuous family of unitaries W = (Wt)tERin 'FI is called a multiplicative 1 - S - cocycle if Wt+s = WtStWsS-t, s, t E R, Wo = I. The cocycle W is said to be a multiplicative 1-S-coboundary if there exists a unitary operator J defining W by the formula Wt =

JStJ*S-t,t E R. Every multiplicative cocycle W determines a new unitary group U = (Ut)tEwin 'H by the formula Ut = WtSt, t E R. This group can be named a cocycle perturbation of S . Notice that if W is a coboundary, then WtSt = J S t J * , t E R, i.e. the coboundary determines the perturbation which is unitary equivalent to the initial group. Consider acting in the Hilbert space 'H = L2(R) the group of shifts S = (St)tEw by the formula (Stf ) ( x ) = f (x t), f E 'H. The group of cohomologies H1(S, L2(R)) is generated by a n additive 1-S-cocycle x = (xt)tGrwdefined by xt(x) = 1, -t < x 5 0, xt(x) = 0 otherwise. The cocycle x satisfies the characteristic properties xt+, = xt +Stx,, I Ixt -x8II = 2It - s11/2, t, s E R. Let 'Ht be a subspace of 'H generated by all functions with supports belonging to the segment [-t, +m). Notice that 'Ht = St'Ho, t E R. We shall call the multiplicative cocycle W Markovian if Wt1xe7it = I, t 0. This property means that W doesn't perturbe "a future" of the system. The Markov property for perturbations was introduced in l . Using the cocycle property W-t = St W,*S-t, t E R, we can rewrite the condition of Markovianity in the form W-tIKeX, = I, t 2 0. It guarantees that linear oper0, are correctly defined and form a Co-semigroup ators & = U-tlHo, t of isometries V in the Hilbert space 'HO.We call V a semigroup associated with the Markovian cocycle perturbation of S . Given a Co-semigroup of isometrical operators V in the Hilbert space 'Ho, one can define the Wold on the subspace 'H(O) reducing V to the decomposition 'Ho = 'H(O)@ semigroup of unitary operators and the subspace 'H(l) reducing V to the semigroup of completely non-unitary operators isomorphic to the semiflow of right shifts in K @ L2(R+), where K: is a Hilbert space with the dimension equal to the deficiency index of the generator of V (see 17). The semigroups Vlx(o, and VIH(~,can be named a unitary part and a shift part of the semigroup V correspondingly. Proposition 2.1. The deficiency index of the generator of the semigroup V associated with the Markovian cocycle perturbation of S equals 1. Proof. Consider a family of functions Ct = W-tx-t, t 2 0, where x is the additive 1 - S-cocycle defined at the beginning of the section. It follows from the definitions of x and W that the family ( is continuous. Then o 1 - V-cocycle. Notice that (KnCt,Ct) = (W-tS-t&(n-l)
+

>

>

+

+

+

( S - t & (. n - l.) C t , ~ - t ) = 0 , n E +m

C eLtn&,ct

n=O

N, t > 0. Therefore the sum

Jt

=

is well defined and we can write the integral sum J = lim tt = t-+O

+m

e-tdCt. It follows that V,*J = eUtJ. So we have proved that the de0

ficiency index of the generator of V is more or equal to one. Let X C be a subspace of 7-l generated by Us O W - ~ (8 ' H%-t) ~ is generated by the see that the subspace functions Ct = WW-tx-tlt 0 , which form an additive 1 - V-cocycle with the orthogonal increaments (see the proof of Proposition 2.1). It follows that the restriction of V to %(') is unitary equivalent to the semiflow of right shifts in L2(R). Let V be a subspace of X o invariant under the action of the semigroup of right shifts S' = S* IR0. Then (see 17) V = M@Ho, where 3 - l M s 3 is the operator of multiplication by an inner function O, 3 is the Fourier transform. Notice that Me is an isometrical operator. Denote P[ott1and PLt,+,) the orthogonal projections on the spaces 'Flo 8 % - t and %-t, t 2 0 , correspondingly. Given a unitary Co-group R = (Rt)tEwin the Hilbert space No 8 V , define a family of unitary operators in %o by the formula

+

+

>

>

In the following proposition we introduce the model describing all Markovian cocycles up to unitary equivalence of perturbations. To be exact, for every Markovian cocycle w there exists the Markovian cocycle W of the form given in the proposition such that Wt = JtWt, t E R, where the 1 - U-coboundary Jt = JWtStJ*W-tS-t, t E R, of the group U = (WtSt)tEwis defined by the unitary operator J satisfying the relation wtst = JWtStJ*, t E R. Proposition 2.3. The family W given by (2.1) i n 'Ho and acting identically in 'H 0 'Ho defines a multiplicative Markovian cocycle such that lim W-,f = M s f for f E 3-10 and lim W-tf = f for f E 'He'Ho. t-++ca t-++m The unitary part of the semigroup V associated with the Markovian cocycle perturbation by W is R. Proof. Extend the family W defined in (2.1) for t L 0 by the formula Wt = StW?,S-t, t 0. Consider the set of unitary operators Ut = WtSt, t E R. Notice that U-~MQ = MQS-t, U-tf E V, f E 'Ht, t 2 0. Hence the subspace L = V @ ((IH 8 ?lo) is invariant under action of Ut, t E R, and the restriction UIL is unitarily equivalent to the group S . To complete the proof notice that the restriction U to the subspace 'HoeV = 'HeC coincides with R*. Theorem 2.4. Given a unitary group which is uniformly continuous or has a pure point spectrum, there exist the inner function G and the unitary group R in the Hilbert space 'Ho 8 MQ'Ho unitarily equivalent to R such that the Markovian cocycle (2.1) satisfies the condition Wt - I E sz, t E R. Theorem 2.4 for the case of uniformly continuous R can be found in 4,5. The proof for R with a pure point spectrum is also constructed in cited papers in the implicit form. One can compare the condition W- I E sz on the unitary operator W appearing in the theorem with the Feldman criterion on the equivalence of Gaussian measures (see '',I3) and the Araki criterion of the quasi-equivalence for quasifree states of the algebra of canonical commutation relations (see '). It seems that the unitary operators W satisfying our condition translate equivalent states one to another. It is also useful to notice that the condition Wt - I E s2, t E R, is nessesary and sufficient for the family W = (Wt)tEw to define an inner cocycle on the hyperfinite factor generated by a quasifree representation of the algebra of canonical anticommutation relations (see 5,16).

>

3. Stationary quantum stochastic process as *-homomorphism into a ring of cohomologies. Let K be an involutive algebra and Ki = KBi. Supply the family (Ki)Ly with a linear associative operation defining left and right actions of every x E Ki on K j such that x @ y, y @ x E Ki+j, y E Kj. We assume that Ki @ K j = Ki+j, i , j E N. So we obtained the graded algebra K = @&Ki with respect to the multiplication defined by the operation a.If K is a Lie algebra, it is possible to take the universal enveloping algebra for K and the multiplication in K for 0. Consider a one-parameter w*continuous group of *-automorphisms a = (at)tEw on the algebra B(H) of all bounded operators in a Hilbert space 'FI. Suppose that the action of a can be correctly defined on certain involutive algebra M consisting of linear operators (in general unbounded) in 'FI. Consider the standard resolvent for a constructed from non-homogeneous chains such that (see 14)

+

+

where di(x)(tl,.. . ,ti+l) = atl( ~ ( t z. ,. . , ti+l)) - ~ ( t l t2,. . . ,ti+l) . . . (-l)ix(tl,. . . , t i ) , x = x(t1,. . . , t i ) E Hom(Ri, M ) . Denote Ai = kerdi/Imdi-l, then A = @:=?Ai is a ring with respect to the multiplication defined by a bilinear map (x, y) 3 x(t1,. . . ,ti)at,+,,,+ti(y(ti+ll... , ti+j)) E Ai+j1 x €Ail y € A j . Every *-homomorphism j of the graded algebra K into the graded algebra A such that every x E Ki is translated into an i - a-cocycle j(x) we shall call a stationary quantum stochastic process over the algebra K . Notice that our definition is based on the well-known one given in 2 . It is also useful to remark that we don't need 0 - a-cohomologies in our construction. Sometimes we can recognize two processes j and determining the cocycles j(x) and j(x) which differ on the coboundary for the fixed x as obtaining one from other by a shift in time. For example, given a stationary quantum stochastic process j the 1 - a-cocycle j(x)(t) is differ on the coboundary at(j(x)(r)) - j(x)(r) from the 1 - a-cocycle j (x)(r t) - j(x)(r) which is associated with the stationary quantum stochastic process ? obtained from j by a shift in time on r. In applications we claim that j keeps the basic algebraic structure in K and the multiplication in K but we do not need to require for j the preserving of the multiplication in K if it is defined (see the next section). We shall suppose that the operators involved in the image of j generate whole M. Proposition 3.1. Let j be defined on K = K1. Then there exists a

+

3

+

unique extension of j to whole K. Proof. Notice that j translates the operation in the cohomological multiplication U. Denote j(i)= jlKi,then one can obtain the action of j by the induction,

for all x E K k , y E Kl. Here we use the property

K iO Kj = Ki+j.

4. Stationary quantum stochastic processes on the algebras of canonical commutation relations, the square of white noise relations and canonical anticommutation relations. Consider an involutive Lie algebra K generated by the elements B, B + , A, 1satisfying the relations [B, Bf] = 1, [A,B] = -B, [A, B+] = B f . Let the operation be generated by the multiplication in the universal enveloping algebra of K . One can define a stationary quantum stochastic process j over K by the formula

j(A)

= At,

j(1) = t l ,

where bt, bf, At are the bosonic anihilation, creation and number of particles processes (see 15). Analogously, if K is the Lie algebra generated by the elements B-, B f , M satisfying the relations of s12 which are [B-, B f ] = MI [M, B*] = f2 B f , one can consider the universal enveloping algebra of K and we obtain the quantum Levy process generated by the square of white noise (SWN) constructed in 3 ,

j(M)

= yt

+ nt,

where the basic processes bt, b,f and nt satisfy the relations of SWN, btb,f - b t b t = yt

+ nt, ntbt - btnt = -2bt,

with a fixed parameter y > 0 and t E R.

The algebra of canonical anticommutation relations ( C A R ) is generated by elements a t , a:, t E R , satisfying the relations ataz +a,*at = t A s l , atas asat = 0. A graded algebra K = $ z 1 K i can be obtained in the following way: K i is a tensor product of i-th copies of the algebra of 2 x 2-matrix units. Let elements ak, a;, k E N,satisfy the canonical anticommutation relations aka;+a;ak = d k l l 1 akal+alak = 0 , k , 1 E N . Then K i is generated by ak,a;, 1 5 k 5 i. Determine a canonical operation o by the formula XI 2 2 o . . . @ x , = y1 y2 . . . ynl where yi = ail af , afai, aiaf if xi = a , a*, a*a, aa* correspondingly. Then a stationary quantum stochastic process j can be defined by the formula

+

j(a*a) = At, j ( 1 ) = t l , where a t , a f , A t are the basic Fermion processes (see 6 ) . Notice that j satisfies the relations j(a*) = j(a)*, j(a*a aa*) = j(a)*j ( a ) j ( a ) j ( a ) * = j ( l ) , j([a*a,a])= [ j ( a * a ) , j ( a )= ] - j ( a ) = at, j([a*a,a*l)= [j(a*a),j(a*)] = j(a*) = a f , but it is not the algebraic *-morphism because j(a*a) # j ( a ) * j ( a ) .

+

+

5. Cocycle perturbations of K-flows and the Wold

decomposition. We shall use the notation of previous parts of this paper. Remember that a of unitary operators in 'FI is named strong continuous family W = (Wt)tEw a multiplicative a-cocycle if Wt+s= Wtat(Ws),s , t E R . Suppose that the action of W is correctly defined on M , i.e. WtxW,* are well defined for all t E R, x E M . Let M t l , M I Sand M [ s , t lbe involutive subalgebras of M generated by all increaments of the form j ( x ) ( r ) - j ( x ) ( l ) , x E K1, where 1 5 r 5 t , s 5 1 5 r and s 5 1 5 r 5 t correspondingly. We shall call (see also I ) a multiplicative a-cocycle Markovian (with respect to the stationary quantum stochastic process j ) if WtMtlW,* c M t l and WtxW,* = x for all x E M [ , , t 2 0. Theorem 5.1. For any Markovian cocycle W the formula

defines a new stationary quantum stochastic process :! over K with an associated group of automorphisms &. Proof. Analogously to the proof of Proposition 2.1 we obtain

s, t 5 0. Here we used the identity at(Ws)j(x)(t)at(W,*) = j ( x ) ( t ) due to the Markovian property Wsa-t(j(x)(t))W,*= -a,(W:,) j(x)(-t)aS( W W s )= -as(W:s(j(x)(-t-s) - j ( x ) ( - ~ ) ) W - = ~ )- a s ( j ( x ) ( - t - s ) - j ( x ) ( - s ) ) = - j ( x ) ( - t ) = a-t ( j ( x ) ( t ) ) , - st ,5 0. One can extend ? ( x ) ( t )for t 2 0 using the cocycle condition for j ( x ) ( t ) . It yields j ( s ) ( t ) = j ( x ) ( t ) , t 0. To complete the proof we only need to apply Proposition 3.1. Proposition 5.2.

>

Proof. From the Markovian property of W immediately follows that &t(x)= Wtat(x)W,* = a t ( x ) , x E M p , t 2 0. The Markovian property 0, which is equivalent to implies that WtxW,* = x, x E Mit, t a t ( W - t ) * x ~ ~ ( W=- x~,) x E M [ t , t 0, or W*txW-t = x , x E M[O, t 2 0, by the cocycle condition for W . Hence & ( x ) = W-ta-t ( x ) W : ~= a-t(x), x E M p , t 2 0. Denote N = M" n B ( H ) , N,] = M ; n B ( H ) , Np = Mi: n B ( H ) and N[s,tl= Mi:,,] n B ( H ) the corresponding von Neumann algebras. Notice that Nt+,] = a t ( N s l ) ,t , s E R. We shall call a stationary quantum stochastic process j a K-flow and the group a associated with j a group of automorphisms associated with the K-flow if the following conditions hold,

>

>

r\tEwNtl = C1 (see l l ) . Proposition 5.3. If there exists a vector R E 'H which is cyclic and separating with respect to JV and the increaments of a stationay quantum stochastic process j are independent i n the classical (commutative) sence that $(xy)= $(x)$(y), xE y E Ntl, t E R, for the state $(.) = ( R ,.R), then j is a K-flow. Proof. Choose x E then $((x- $ ( x ) l ) y = ) $(a:- $ ( x ) l ) $ ( y= ) 0 for = vtGwNt1 = N . Hence (x - $ ( x ) l ) R= 0 and x = $ ( x ) l all y E vtEwNit as R is cyclic and separating. The result follows from. Let von Neumann algebras Gtl, and be associated with the are perturbed process 3 in the same way as the algebras Ntl, NIs,tland associated with the process j . Proposition 5.4. Let j and W be a K-flow and a Markovian cocycle correspondingly. Then the Markovian perturbation ? is also K-flow. Proof. One can see that the von Neumann algebras generated by the increaments of are Ntl = WtNtlW,*c Ntl by the Markovian property. Hence = C1. The conditions

4,

qs,tl qt

Nit

3

are satisfied by the definition. For a stationary quantum stochastic process j one can name the Eoassociated semigroup Pt = a-t lNol, t 2 0 , on the von Neumann algebra with j. In the case when j is a K-flow, the semigroup ,B = ( P t ) t 2 0 is a flow of Powers shifts 1 8 , i.e. AnE~Ptn(NOl) = C1, t > 0 (see '). Fix a stationary quantum stochastic process j with the associated Eo-semigroup p. For a cocycle W being Markovian with respect to j we shall call the Eo-semigroup fit(.) = W-tj3t(.)W?t,t 0 , on No] associated with the Markovian perturbation of the initial process by W . Two Eo-semigroups P and f i on the von Neumann algebras A and A correspondingly are called to be conjugate if there exist two injective *-homomorphisms 8 : A -+ A and B+ : A + A such that Pt(x) = 8+ptB(x),e+e(x)= X , 88+(y) = y , x E A, y E A, t L 0. Theorem 5.5. Given a Markovian perturbation of the K-flow j with the acting associated flow of Powers shijls ,6' on the von Neumann algebra

>

i n the Hilbert space 3-t with a ciclyc vector 0 , there exists the von Neumann algebra c No] such that the restriction of the Eo-semigroup 6 associated with the Markovian perturbation is conjugate to P. Proof. P u t I f t = [N[-tO], t E R, then the set I f t , t 0 , is dense in the Hilbert space 3-t. Arguing similarily to the proof of theorem 2.2 one can obtain that there exists s - lim WPt = W-,. Then the t++m injective *-endomorphism 19 : No] -' No] c No] given by the formula B(x) = W-txW_Tt, x E Nl-t,ol, t 0 , is well defined because W-t-sxW*t-s = W-~~-~(W-,)X~_~(W*,)W_T, = W - t x W r t for all x E N[-t,olby the Markovian property W-,yW:, = y, y E N[O,which implies that W-,(rt(x)W_T, = a t ( x ) for all x E Nl-t,ol. It follows that lim W - t x W I t f = lim W - t x W 2 , f = W-,xW:, f for all f E 3-1-,, x E t-++a t++m No],s 0. Hence the sequence W - t x W 2 t f converges when t tends t o +cc for all f E 'Hal by the Banach-Steinhaus theorem. Analogously it is possible to define the injective *-homomorphism B+ : No]-' No]by the x E q - t , o l , t > 0. Notice that B+(x) = formula B+(x) = W_T,XW_~, W_T,xW-,, x E N o ] . One can see that P t ( x ) = BPtB+(x), x E No].This proves the theorem. Earlier it was investigated the existence of "an automorphic part" in the quantum dynamical semigroup which is completely compatible with the faithful normal state (see l o ) . Theorem 5.5 allows t o obtain " a shift part" of the Eo-semigroup obtained by a Markovian cocycle perturbation from the flow of Powers shifts. So it can be considered as some analogue of the picking out a completely non-deterministic part in the Wold decomposition for the classical stochastic processes.

PIfio,

>

>

>

Acknowledgements The author is grateful to Professor Luigi Accardi for kind hospitality during his visit a t Centro Vito Volterra Universita di Roma Tor Vergata where a part of this work was done.

References 1. L. Accardi, Rendiconti del Seminario Matematico e Fisico, Milano 48, 135180 (1978). 2. L. Accardi, A. Frigerio, J.T. Lewis, Publ. R.I.M.S. Kyoto Univ. 18, 97-133 (1982).

3. L. Accardi, U. Franz, M. Skeide, Centro Vito Volterra Universita di Roma Tor Vergata, Preprint 423 (2000). 4. G.G. Amosov, Izv. Vysch. Uchebn. Zaved. Matem. 2,7-12 (2000). 5. G.G. Amosov, Infinite Dimensional Analysis, Quantum Probability and Rel. Top. 3,237-246 (2000). 6. D. Applebaum, R. Hudson, Commun. Math. Phys. 96,473-496 (1984). 7. H. Araki, S. Yamagami, Publ. R.I.M.S. Kyoto Univ. 18, 283-338 (1982). 8. B.V.R. Bhat, Memoirs of the AMS 709 (2001). 9. A.V. Bulinskij, Russ. Math. Surveys 51,321-323 (1996). 10. A.V. Bulinskij, Funk. Anal. Pril. (Funct. Anal. Appl.) 29,64-67 (1995). 11. G.G. Emch, Commun. Math. Phys. 49,191-215 (1976). 12. J. Feldman, Pacific J. Math. 8, 699-708 (1958). 13. A. Guichardet, Symmetric Hilbert spaces and related topics (Springer Lecture Notes in Mathematics 261, 1972). 14. A. Guichardet, Cohomologie des groupes topologiques et des algebres de Lie (Paris, 1980). 15. R. Hudson, K.R. Parthasarathy, Commun. Math. Phys. 93,301-323 (1984). 16. T. Murakami, S. Yamagami, Publ. R.I.M.S. Kyoto Univ. 31,33-44 (1995). 17. N.K. Nikolski, Treatise on the shift operator (Springer, 1986). 18. R.T. Powers, Canad. J. Math. 40,86-114 (1988).

THE STOCHASTIC LIMIT AND THE QUANTUM HALL EFFECT: ELECTRONS AND QUONS

FABIO BAGARELLO Dipartimento di Matematica ed Applicazioni, Fac.Ingegneria, Universita di Palermo, I - 90128 Palermo, Italy E-mail: [email protected] We review here some recent work by L. Accardi and the author, concerning an alternative explanation of the experimental results concerning the resistivity tensor for a (almost) two-dimensional electron gas (2DEG) using the perturbative approach provided by the so-called stochastic limit. We also discuss the extention of this approach to a gas of quons.

1. Introduction In this paper we discuss some dynamical facts concerning a physical model defined by an Hamiltonian H = H0,e + H0
(I)

which is obtained from the Hamiltonian of the quantum Hall effect (QHE)1'4'5, by replacing the Coulomb background-background interaction by the free bosons Hamiltonian HQ^R, see (10) below, and the Coulomb electron-electron and electron-background interaction by the Frohlich Hamiltonian Heb (13) which is only quadratic rather than quartic in the fermionic operators. However, since from the Prohlich Hamiltonian it is possible, with a canonical transformation, to recover a quartic interaction 3, we can say that the Prohlich Hamiltonian describes an effective electron-electron interaction which may mimic at least some aspects of the original Coulomb interaction. From this point of view it seems natural to conjecture that some dynamical phenomena deduced from this Hamiltonian might be relevant in the study of the real QHE. This conjecture is supported by our main result, given by formulae (70) and (71) where we deduce, directly from the dynamics, an obstruction to the presence of a non zero ^-component of the current, which is quantized according to the values of a finite set of rational numbers. This is a new

41

42

and exciting result and it is surely worth of some attention. More precisely, we prove that the x-component of the mean value of the density current operator is necessarily zero unless a certain quotient (^lf , cf. (7) and (9) for the definition of these parameters), takes a rational value. This is what we call a fine tuning condition (FTC). Even if there is no mathematical limitation for the values of these rational numbers, it is quite reasonable to expect that, in a concrete physical situation, only a small set of numbers will play a relevant role for the scale of phenomena involved. Of course the FTC strongly reminds the rational values of the filling factor for which the plateaux are observed in the real QHE. We will discuss all these facts and we will also show that if we replace the gas of electrons with a gas of quons, 6789 ' ' ' , the FTC still appear. It is clear that the use of quons in QHE is, in a sense, quite natural from a phenomenological point of view since it is well known that strange particles appear in the game. These excitations are usually called anyons in the literature. We use the technique of the stochastic limit of quantum theory and we refer to the paper 10 for a synthetic description, to u for more recent results, to 12 for mathematical details and to 13 for a systematic exposition. 2. The description of the model In this section we specify the various components of the model discussed in the Introduction. We begin with briefly reviewing the single electron problem and then we give the second quantized version of the hamiltonian in (1).

2.1. The single electron problem In these notes we discuss a model of N < oo charged interacting particles concentrated around a two dimensional layer contained in the (x, y)-plane and subjected to a uniform electric field E_ = Ej, along y, and to an uniform magnetic field B_ — Bk along z. The Hamiltonian for the free N electrons HQ ' , is the sum of N contributions: (0

(2)

where HQ(i) describes the minimal coupling of the i-th electrons with the field: eE.r_i

(3)

43

To HQ we have to add the interaction with the background and, then, the free Hamiltonian for the background itself. This will considered below. We fix the Landau gauge A = —B(y,0,0). In this gauge the Hamiltonian becomes 9

~

which, obeys the commutation rule \px,Ho] = 0. The solutions of the eigenvalue equation for the single charge Hamiltonian (4) H0ipnp(r) = enpif}np(r),

n e N, p e Z

(5)

(where the double index is due to the fact that, two quantum numbers are necessary to fix the eigenstate) are known, 5, to be of the form: ip(r) = Celkx
1 , 1

„

(6)

depending on the parameters pR -° C

'

2 h/ I2]cft.

.

11

0..2

.

„

1 L

/tL, .

^77.\

.

where k is the momentum along the x-axis. If we require periodic boundary condition on x, i/j(—Lx/2,y) — il>(Lx/2,y}, for almost all y, we also conclude that the momentum k along x, cannot take arbitrary values but must be quantized. In particular, if the system is infinitely extended along y, then all the possible values of k are:

k = ~P, L/x

PeZ

(8)

Normalizing the wave functions in the strip [— Lx/2, Lx/2] x R, we finally get:

(9) where
44

Equation (9) shows that the wave function VVipfc) factorizes in a x— dependent part, which is labelled by the quantum number p, and a part, only depending on y, which is labelled by both n and p due to the presence of 2/Q in the argument of the function (pn . It may be interesting to remark that when E — 0 the model collapses to the one of a simple harmonic oscillator, see 5 and 1 for instance, and an infinite degeneracy in p of each Landau level (n fixed) appears. Following the usual terminology we will call lowest Landau level (LLL) the energy level corresponding to n = 0. 2.2. The second quantized model The Hamiltonian HQ contains the interaction of the electrons with the external electric and the magnetic field. In this paper we add the Frohlich interaction of the electrons with a background bosonic field. The free Hamiltonian of the background boson field is HO,R =

u(k)b+(k)b(k)dk

(10)

where ui(k) is the dispersion for the free background. Its analytical form will be kept general in this paper. The electron-background interaction is given here by the Frohlich Hamiltonian 3 Heb = /V(rMr)F(r - r')<j>(r')drdr'

(11)

where ip(r) and <j>(r?) are respectively the electron and the bosonic fields, while F is a form factor. Expanding (j)(r) in plane waves, tp(L) in terms of the eigenstates ^a(L)j see (9), introducing the form factors 9a0(k)

:= — L=3 -^^ where Va(i,(k) := / V«(r)e**lMr)dr x/(27r) J /(27T) \/2uj(k)

and taking F(r_) = e26(r_), 3, we can write Heb = e2Y^^a0(b(gaft)

+ b+(g^))

(13)

a/3

which is quadratic in the fermionic operators aa, a+, {aa,a/j} = {a+,a^} =0

{aa,oj} = 5a0

(14)

The boson operators b(k) satisfy the canonical comutation relations: [b(k),b+(k')}=5(k-k')

(b(k),b(k')} = [b+(k),b+(k')} = 0

(15)

45

The form factors gap depend on the level indices (a,/3). Notice that we have adopted here and in the following the simplifying notation for the quantum numbers a = (na,pa) and that we have introduced the smeared operators a)

= jdkb(k)9f}a(k}.

(16)

In terms of the fermion operators, the free electron Hamiltonian (2) becomes:

where the ea are the single electron energies, labeled by the pairs a = (n,p), see formula (9). Therefore the total Hamiltonian is: H = H0,e + HO,R + \Heb = H0 + XHeb

(18)

3. The electron gas and the stochastic limit In this section we briefly outline how to apply the stochastic limit procedure to the model introduced above. We will consider the gas of quons in the next section. The starting point is the Hamiltonian (18) together with the commutation relations (14), (15). Of course, the Fermi and the Bose operators commute among them. The interaction Hamiltonian Heb for this model is given by (13) and the free Hamiltonian HQ is given by (17), (10) and (18). The time evolution of Heb, in the interaction picture is then Heb(t) = eiHotHAe-iHot = e a/3

(19)

where

£a/3 = e<* - e/3

(20)

Therefore the Schrodinger equation in interaction representation is: dtU™ = -i\Heb(t)U^

(21)

After the time rescaling t —» t / X 2 , equation (21) becomes

U$l

(22)

46 whose integral form is

We see that the rescaled Hamiltonian \ Heb(t/X2) = e2 V>U4 b (e^ (w - ea0)ga0] + h.c.

(24)

a/3

depends on the rescaled fields

A The first statement of the stochastic golden rule see 13 is that the rescaled fields converge (in the sense of correlators) to a quantum white noise baB\f)

=

hiri — b(gage ~>2 <*0>\ A-»o A characterized by the following commutation relations

/j.f\i at \L

c Of

)

Jf/-/• a

f

/

/ 0 \L

(26)

-f'N /^OL0ot B C

1^-J

/OQ\ I ^O )

where the constants Gal3a @ are given by f-iapa'0' \jr

_ —

r°° / J — 00

r

j LLI

, ,

%

I jr. a(1f\n ,„ /L\.,iT(u;(fc)-e a/ 9) _ t tA/iy^/J 1 /v JycK 3 \~) — J

= 27T / " d k g a p ( k ) g a p ( k } 5 ( w ( k )

- 6af})

(29)

The vacuum of the master fields 6^/3 (i) will be denoted by 770, ba/3(t)rj0=0

Va/3, Vt

(30)

13

The limit Hamiltonian is, then, see . a/3

In this sense we say that H^, (t) is the "stochastic limit" of Heb(t) in (19). Moreover, the stochastic limit of the equation of motion is (22) dtUt = -iH™(t)Ut

(32)

Ut=TL-i I H(esbl\t')Ut,dt' Jo

(33)

or, in integral form,

47

Finally, the stochastic limit of the (Heisenberg) time evolution of any observable X of the system is: jt(X) = U+XUt = U+(X lR)Ut

(34)

Since the bap(t) are quantum white noises, equation (32), and the corresponding differential equation for j t ( X ) , are singular equations and to give them a meaning we bring them in normal form. This normally ordered evolution equation is called the quantum Langevin equation. Its explicit form is: dtjt(X) = e2 ^{jt((a+a0,X}ra_0 - raJ3[a+aa,X})}+

(35) a/3

where pa/3

V~^ r

+

/og 1 )

;£,a/3a'/3'

a'/3' O

/ 1

=

,-

cir / dkgap(k)ga'f3'(k)eiT(aj(-^~e'"3} -OO

/ / /~YQtj3ot (3

2

=

(37)

«/

' T} TJ

~

/" /

/IA \*^)

(."'J

/ ^a/3^a'/3'

1

Z

The master equation is obtained by taking the mean value of (35) in the state TJQ = rj0 £, £ being a generic vector of the system. This gives a/3

and from this we find for the generator o/3«'/3'

^

°

(39)

The expression for L(X) obtained above will be the starting point for our successive analysis, which begins with the definition of the current density operator. The current is proportional to the sum of the velocities of each electron: ^oV^-RiW.

(40)

48

Here A is the two-dimensional region corresponding to the physical layer, ac is a proportionality constant which takes into account the electron charge, the area of the surface of the physical device and other physical quantities, and Ri(t) is the position operator for the i-th electron. Moreover N is the number of electrons contained in A. Defining N

(41) we conclude that (42)

Since X\(t) is a sum of single-electron operators its expression in second quantization is given by

'

a+a

(43)

where

= /

(44)

Recall that the V'-y(n) are the single electron wave functions given by (9) and aa and a+ satisfy the anticommutation relations (14). We refer to 2 for the computation of the matrix elements (44). Here we give only the result which is A

L-rC

p

(45)

-:) (46)

where (47)

(48) Notice that, whenever as: X-tu = 0.

= p7, formula (45) must be interpreted simply

49

To show how these results can be useful in the computation of the electron current we start noticing that, if Q is a state of the electron system, then (/ A (0) e = <*c(jt *A(*)>* = a c (L(* A (t))) e = acTr(0L(^(t)))

(49)

The vector (JA.(t))e will be now computed for a particular class of states g, and we will use this result to get the expressions for the conductivity tensor and for its inverse, the resistivity matrix. To do this we begin computing the electric current. We first need to find L(X\), L being the generator given in (39). Since XA. = -^1> we have where, as we find after a few computations, ,

In the present paper we consider a situation of zero temperature and we compute the mean value of LI(XA) on a Fock ./V-particle state ipr. •0/ =a£ ...a^Vo,

ik^ii,^k^l

(51)

where 7 is a set of possible quantum numbers (7 c (N 0 ,Z)), Nj is the number of elements in 7 and tpo is the vacuum vector of the fermionic operators, aail>0 = 0 for all a. The details of the computation are contained in 2 where we prove that the average current is proportional to >«,=£i(A)+£2(A) and we have isolated two contributions with different structure:

(52)

=e 2 a/3a'

a/3/3'

Using equations (45), (46) for Xfy we are able to obtain £.i(X^') and £i(X^ ), i = 1,2. First of all we can show that, even if Ci(X^') is not zero, nevertheless it does not depend on the electric field. Therefore f fy{)\ n OELl(** > - °

ft^\

(55)

50

Secondly, the computation of C^X^ ) gives rise to an interesting phenomenon: due to the definition of X$, the sum in (54) is different from zero only if p/3 ^ pp>. Moreover, we also must have ep = ep<, that is np - np, =

(pp, - pp)

(56)

This equality can be satisfied in two different ways: let us denote 72. the set of all possible quotients of the form (np — np>)/(pp' — pp). This set, in principle, coincides with the set of the rational numbers. Therefore 0 e 72.. Then 1) if ^ff^ is not in 72, (56) can be satisfied only if /? = /?'. But this condition implies in particular that pp = pp>, and we know already that whenever this condition holds, then Xppr = 0, so that 2) If J^ff again

is in 72., then we have two possibilities: the first one is

which, as we have just shown, does not contribute to £2(^1 )• The second is np - np> _ -

P0' — Pp

which gives a non trivial contribution to the current. Therefore, we can state the following

PROPOSITION. In the context of Model (18) there exists a set of rational numbers 72. with the following property: if the electric and the magnetic fields are such that if the quotient

does not belong to 72. then

51

On the other hand, if condition (57) is satisfied, we can conclude that the sum ^2aggi ^€l3,tl3i (• • •} in (54) can be replaced by

a/3/3'

a

(3fj'

means

where J^a S/3/3 that the sum is extended to all the a and to those J3 and /?' with pp ^ pp> satisfying (57) (which automatically implies that e/3 = e/3')_ Since, as it is easily seen, gap(k)ga'/3'(k) does not depend on E, we find that 9

a

a''

•

he

- . . ' S '

(59)

/rO\

where ffjVff

=

dr f dkgap(k)galli,(k)eir^-^

I J — 00

(60)

J

so that, using also (58), we get

where a

BP'

and

,g(1)/ _ j jfW,

/£ R_)

(53)

Therefore we conclude that

Let us now compute the second component of the average current: The first contribution is easily shown, from (53) and (46), to be identically zero, since dea,ff>Spc.P(i

— <>a/3

(65)

On the contrary the second term, £-2(X^ ), is different from zero and it has an interesting expression: in fact, due to the factor Spfiip^, the only

52

non trivial contributions in the sum Y^pp 8ef}tf ,, in (54), are exactly those with (3 = (3'. Taking all this into account, we find that = =

- y ) X l ( a ) ( l - X/(/?))(GQ/ + G )

e2

(66)

a/3

which is different from zero. Furthermore, using (59), we get

were we have defined ©y - £(Pa - P/3) 2 X/(a)(l - X/03)) Im (A«^)

(67)

a,/3

and Al0a/3 is given by (60). If we call now dE

|t=0 =

=

c

BE

we obtain the conductivity tensor (see 4 ) GXX = Vyy = jy,B,

^xy = -ffyx

= jx,E

(68)

and the resistivity tensor yy

x

y

yy

%y

After minor computations we conclude that Pxy = {

muLx

__ 1

__

6^

aceH e»

™ 2treE

:

g

27reg

^

(70)

-p

("I)

We want to relate these results with the experimental graphs concerning the components of the resistivity tensor, see 5. To avoid confusions, let us remark that our choice for the direction of the electric field, the y axis, is not the usual one, the x axis, see 5. Therefore, in our notation, the Hall resistivity is really pxx, while our pxy corresponds to the xx component of p as given in5.

53

Let us now comment these results which are consequences of the basic relation (57). As it is evident from the formula above, the fact that the fine tuning condition (FTC) (^lf G 7£) is satisfied implies that pxy ^ 0, so that the resistivity tensor is non-diagonal. Vice-versa, if the FTC is not satisfied, then p = pxxTL, I being the 2 x 2 identity matrix. This implies that, whenever the FTC holds, then the ^-component of the mean value of the density current operator is in general different from zero, while it is necessarely zero if the FTC is not satisfied. If the physical system is prepared in such a way that ^ff € ft, then an experimental device should be able to measure a non zero current along the x-axis. Otherwise, this current should be zero whenever J^l^ ^ 7£. A crucial point is now the determination of the set 7£, of rational numbers. ^From a mathematical point of view, all the natural integers na and all the relative integer pa are allowed. However physics restricts the experimentally relevant values to a rather small set. In fact eigenstates corresponding to high values of na and pa are energetically not favoured because the associated eigenenergies enaPc., in (9) increases and the probabilities of finding an electron in the corresponding eigenstate decrease (this is a generalization of the standard argument which restrict the analysis of the fractional QHE to the first few Landau levels). Moreover, high positive values of — pa are not compatible with the fact that HQ must be bounded from below, to be a 'honest' Hamiltonian. Therefore, in formula (57) not all the rational numbers are physically allowed but only those compatible with the above constraints. For this reason it is quite reasonable to expect that the set "R. consists only of a finite set of rational values. The determination of this set strongly depends on the physics of the experimental setting and we shall discuss it in a future paper. We end this section with the following two remarks: the sharp values of the magnetic field involved in the FTC may be a consequence of the approximation intrinsic in the stochastic limit procedure, which consists in taking A —> 0 and t —> oo. In intermediate regions (A > 0 and t < oo), it is not hard to imagine that the S-function giving rise to the FTC becames a smoother function, so that the finite-size plateaux in the plots for py, 5, are recovered. Under special assumptions on the S-dependence of 0X and Qy, together with some reasonable physical constraint on the value of the magnetic field, it is not difficult to check that pxx has plateaux corresponding to the zeros of pxy and that, outside of these plateaux, it grows linearly with B.

54

4. The quon gas and the stochastic limit In this section we show how to apply the stochastic limit procedure to a modified version of the model introduced above, that is to a gas of quons. The physical reason why we are interested in this extension of the previous results is related to the experimental data concerning FQHE: it is a well known fact that fractional excitations appear in the physical sample, that is particles with fractional charge and statistics. Of course, the best we could expect is that the electrons of the (almost) two-dimensional gas change their statistics as a consequence of the dynamics itself when we switch on the magnetic field. This change of statistics for the particles of the system has been theoretically found in some physical model, 14, using the stochastic limit methods. However, this is not an easy task, and here we prefer to consider as a starting point a situation in which the quons already exist. Our aim will be to show that also if the statistics of the system particles is not the usual Fermi one, a FTC is still recovered. This suggests that the FTC is, in a sense, an intrinsic feature of the system, independent of the details of the model, and in this perspective it deserves a deeper analysis. Following the above remarks, we take the hamiltonian exactly as before, (18), the only difference being in the anticommutation relations for the operators aa. We replace the CAR in (14) with the a a a]j - qa^aa = 5a/3,

(72)

q € [—1,1], while no (generalized) commutation rule is imposed on the aa and to the «£, see 8, for instance. For these operators we can still introduce a vacuum, ^o, such that a a ^o = 0 for all a, and a self adjoint number operator for the mode a, na, which is no longer a^aaa because it is clear that allaa(alf)k^o ^ k(alf)k^/Q. The explicit definition of na can be found, for instance, in 8. Here we only want to stress that this operator satisfies the canonical rule (na,a,f3\ = -6a,pa/3.

(73)

With this in mind we have that the free hamiltonian of the electrons in (18), #0,0 is replaced by #o,e = ^,a ^ana, while the other ingredients are formally the same. The procedure for getting the stochastic limit of this model is analogous to that of the previous section. In particular, since Ho<e with our definition satisfies [Ho,e,af}} = —epap, as for ordinary fermions, it is not surprising that the free evolved interaction hamiltonian Heb(t) = elHot Hebe**1*0* formally coincides with the one in (19) so that also the limit hamiltonian

55

coincides formally with the one in (31), Hel (t), the only difference being in the statistics of the quon operators aa. On the other way, the commutation rules for the operators ba^(t] are the same as before, (27), (28). Again 770 is the vacuum of these operators. Pushing on this parallelism with the electron gas, we find that also the form of the two generators formally coincide: also for the quon gas the expression of L(X) is given by (39). Up to now everything looks very close to the previous situation. This is not a case. In fact up to now we have essentially played only with the reservoir, since the change in the definition of HQ^ was such to leave the free dynamics of the quon operators unchanged. Now, in order to compute the current operator and its mean value, the modified commutation rules will play a role, and some difference will eventually appear. The form of the current operators is given by (40)- (48). In order to compute the conductivity and the resistivity tensor we have to compute the commutators like [0^0/3, XA], where we have to use (72), and we must take the mean value of the result on a Fock vector ^tj.ij,.....^ := CWat i at j ....at jv * 0 , < . > e =< ^i^,....,^, .$i1,i2,....,iN >. Here CN is a normalization constant (CN = 1 for fermions), and a difference with respect to the Fermi situation appears: the order in the ik indices above is essential, since we have no rule to exchange creation (or annihilation) quon operators but for q — ±1. In order to simplify the situation we choose, from now on, the deformation parameter q = 0. In this way we get, after some technical computation, the following expression for the mean value of the current: (J A (i)) e = n c a c e 4 { ^ , v ( / 3 / 3 ' G l J / l / + h.c.) /3/3'

5ea,6ti(XiiaGa_

+ h.c.)},

(74)

Doc

with nc being a normalization constant related to the CN above. It is not difficult to check in particular that, whenever the FTC is not satisfied, then the x component of (74) is zero, while it is not zero if £ Ti- So we can conclude that a FTC appears also for the quons. Of course, most of the simplicity in our final computation is due to the choice q — 0. Since the value of q should be given by the experimental data, we believe that an extra effort should be made to generalize our results to any value of q £ [—1, 1]. However, our claim is that this extension will not modify the main result of this section, namely the existence of a FTC, but

56

only the details of the computation. What is more interesting, from our point of view, is considering the harder approach in which the statistics of the system should be modified by its interaction with the reservoir. Further analysis in this direction is in progress. Acknowledgements It is my great pleasure to thank the organizers of the conference for inviting me and for financial support. References 1. F. Bagarello, G. Marchio, F. Strocchi, Phys. Rev. B 48, 5306 (1993). 2. L. Accardi, F. Bagarello, Phys. Lett. A (submitted 2001) 3. F. Strocchi, Elements of quantum mechanics of infinite systems, World Scientific, Singapore-Philadelphia. 4. T. Chakraborty and P. Pietilainen, The FQHE, Springer-Verlag, Berlin, 1988. 5. S.M. Girvin, The Quantum Hall Effect: Novel Excitations and Broken Symmetries, Springer Verlag, 1999. 6. R.N. Mohapatra, Phys. Lett. B 242, 407-411 (1990) 7. D.I. Fivel, Phys. Rev. Lett. 65, 3361-3364 (1990); Erratum, Phys. Rev. Lett. 69, 2020 (1992) 8. O.W. Greenberg, Phys. Rev. D 43, 4111-4120 (1991) 9. M. Bozejko, R. Speicher, Comm. Math. Phys. 137, 519-531 (1991); P.E.T. Jorgensen, L. M. Schmitt, R. F. Werner, Pac. J. Math. 165, 131-151 (1994); D.A. Dubin, M.A. Hennings, A.I. Solomon, J. Math. Phys. 38, 3238-3262 (1997) 10. Accardi L. , S.V. Kozyrev, I.V. Volovich, Phys. Rev. A 3, 56 (1996) 11. L. Accardi, S.V. Kozyrev and I.V. Volovich, Journal of Physics A 32, 34853495 (1999) 12. Accardi L., Frigerio A., Lu Y.G., Comm. Math. Phys. 131, 537-570 (1990) Accardi L., Lu Y.G., Comm. Math. Phys. 180, 605-632 (1996) 13. Accardi L., Y.G. Lu, I. Volovich, Quantum Theory and its Stochastic Limit, Springer, 2001. 14. Accardi L., Lu Y.G., in Quantum Probability and Related Topics Vol. VIII,118 (1993),World Scientific

THE FELLER PROPERTY OF A CLASS OF QUANTUM MARKOV SEMIGROUPS II *

RAFFAELLA CARBONE Universita degli Studi di Pavia, Dipartimento di Matematica "F. Casorati", Via Ferrata 1, 27100 Pavia, Italy FRANCO FAGNOLA Universita degli Studi di Genova, Dipartimento di Matematica, Via Dodecanese 35, 16146 Genova, Italy

Let [) be a Hilbert space and let B(t)) be the von Neumann algebra of all bounded operators on f). We characterise to*-continuous Quantum Markov Semigroups (Tt)t>o enjoying the Feller property with respect to the C'-algebra >C(tj) of compact operators i.e. such that AC(fj) is 7f-invariant and (7t|;c(i,))t>o is astrongly continuous semigroup on /C(h).When (7t)t>o is the minimal Quantum Markov Semigroup associated with quadratic forms ~£(x) (x 6 B(h)) given by -£(x)[u,u] = (Gv, xu) + ^2t(Ltv,xLiu) + (v,xGu) with possibly unbounded operators G, LI we show that the Feller property with respect to /C(h) holds under a summability condition on the L|. We also show that the quantum OrnsteinUhlenbeck semigroup enjoys the Feller property with respect to a bigger C""-algebra including £(h) and functions of position and momentum operators.

1. Introduction

A quantum dynamical semigroup (QDS) on a von Neumann algebra A is a semigroup T = (Tt)t>0 of bounded, normal, completely positive maps on A such that for each a € A, the map t —» 7t(a) is continuous with respect to the weak* topology on A. "THIS RESEARCH HAS BEEN SUPPORTED BY THE MURST PROJECT "QUANTUM PROBABILITY AND INFINITE DIMENSIONAL ANALYSIS 2000-2001".

Keywords and phrases. Quantum dynamical semigroups, Feller property. Mathematics Subject Classification. Primary: 46L55; Secondary: 47N50, 81S25 57

58

A quantum dynamical semigroup is Markov (i.e. it is a QMS) if it is identity-preserving (i.e. Tt(T) = 1 for each t > 0). This is a non-commutative generalisation of the classical notion of Markov semigroup on L°°(E,£,fj1) ((E,£) measurable space, fi a finite measure on E) since a map on a commutative von Neumann algebra is completely positive if and only if it is positive (see [29]). As in the classical theory of stochastic processes it is interesting to study the regularity of a QMS (e.g. to deduce properties of the corresponding Markov process). This note is concerned with the following property Definition 1.1. We say that a QDS T = (Tt)t>0 on a von Neumann algebra A enjoys the Feller property with respect to a sub-C""-algebra AQ of A (or is «40-Feller) if AQ is 7^-invariant for each t > 0 and, moreover, the restriction of the operators Tt to AQ yields a strongly continuous semigroup on AQ. This means that Tt(Ao) C Ao and the map t —•> 7<(a), for a G Ao, is norm continuous i.e. lim^o ||^t(a) — a\\ = 0. (Our terminology here is different form that of our previous paper [7] and closer to that of classical Markov semigroups [13]). This property is useful since several results of semigroup theory (e.g. those related to the notion of dissipativity or those on perturbations) hold for strongly continuous semigroups but not for w*-continuous semigroups. In [7] we studied the Feller property of QMS on £(h) with respect to the C*-algebra /C(h) of compact operators on h. We characterised norm continuous /C(())-Feller QDS and gave some conditions, based on the Ttinvariance of the Banach space 2i(h) of trace class operators on f), for a w*-continuous QDS to be £(h)-Feller. In this paper we start by introducing the Phillips' space of a QDS T. This is the biggest closed subspace X of A on which T is & strongly continuous semigroup. R.S. Phillips ([26]) proved that X is always weak*-dense in A. Here we show (Prop. 3.1) that, for a QMS on B(fy"), it often contains compact operators. However, as it might happen for classical Markov semigroups (Prop. 3.2), it is not always an algebra. Then, regardless of pathologies that appear for both classical and quantum Markov semigroups, we concentrate on the Feller property of QDS on B($) with respect to the C'*-subalgebra /C(f)). We extend our previous characterisation of norm-continuous /C(h)-Feller QDS to arbitrary weak*continuous QDS (Prop. 4.1, Th. 4.1).

59

Moreover, we consider QMS with weak*-infinitesimal generator associated with quadratic forms £(x) (x e B(t))) given by

£(x)[v,u] = (Gv,xu) + y

(LfV,xLfu)

+ (v,xGu)

(1)

with possibly unbounded operators G, L(. In this case we give (Th. 4.2) a quite general sufficient condition based on the operators L| for the minimal QDS to be /C(P))-Feller. This condition has been used also in ([7] Th. 4.1, Th. 4.2) but our proof here is simpler. Our characterisation also applies to semigroups acting on the von Neumann algebra B(\)) where f) is the Hilbert space L2(R.d; C) of the Schrodinger representation of the CCR over Cd (see [5], [25]) and admit a generator like (1) with G, Le depending on (ar,CL*)r=i,...,d, the creation and annihilation operators d

In this case, however, /C(f() seems too small since it does not contain non-trivial functions of position and momentum operators

qr = (a* + a r ) /\/2,

pr = i (a* - ar) /Vl.

Indeed, for all non-zero /, g & C0(]R rf ;C) (the Banach space of continuous functions on Md vanishing at infinity) the operators f(p),g(q) (p = (pi,... ,p exp(i(£,:r)) (£ g R d ). Indeed the corresponding f(p),g(q) are the Weyl operators W(—£/V%), W(il;/^/2) respectively. Moreover, for d — 1, LI — a, G — —(l/2)LJLi, the QMS associated with (1) acts as (see e.g. [2]) Tt(W(z)) = exp (-|z|2 (1 - e-*) /2) W(e-t/2z),

z 6 C.

Therefore limsup t _ 0 \\Tt(W(z)) - W(z}\\ = 2 since the norm \\W(zv) W(zi)\\ is equal to 2 for zi,z2 e C with z\ ^ z2 ([5] Prop. 5.2.4 p.13). The quantum Ornstein-Uhlenbeck semigroup corresponds to the choice d = 1, LI = Aa, L2 = ^a*, Lt = 0 for t > 1, G = -(A 2 /2)Ll-Li - ( f J , / 2 ) L ^ L 2 with A, n > 0. We shall show that this semigroup is also Feller with respect to the bigger C**-algebra M generated by operators f(p),g(q) with /,g e C°(K d ;C) and the identity 1 (In Section 5 we discuss the structure of this algebra). Unfortunately similar semigroups, with G, L( depending in a

60

simple way from (a r ,a*)r=i,...,d are not .M-Feller. Thus, even for these semigroups, M is not a good choice as the C*-algebra of compact operators. In Section 6 we prove that several QMS arising in Quantum Optics are /C([))-Feller. 2. Notations and preliminaries Any QDS is the dual semigroup of a strongly continuous semigroup on the predual of the von Neumann algebra A (see e.g. [14] Ch.3). We recall the basic results of Phillips's theory of dual semigroups referring to [6] Sect 1.4 for a detailed account. The Phillips' space ([26]) of a QDS T is denned as lim \\Tt(x)-x\\ = O *^°+ Right continuity at 0 is equivalent to continuity on [0, +oo[ because the maps Tt are locally uniformly bounded ([4] Prop. 3.1.3 p.164). We refer to [6] Prop. 1.4.6 p.50 and Prop. 1.4.7 (a) p.51 (strongly closed there means closed in the norm of A as a Banach space) for the proof of the following Proposition 2.1. Let T be a QDS on a Von Neumann algebra A. (a) X is a norm-closed, self-adjoint, Tt-invariant, linear submanifold of A, (b) the domain of the w*-generator C is contained in X, (c) X is equal to the norm closure of dom(£.) in ARecall that the w* -generator £ of a QDS T is the operator on A with domain the set of all those x £ A such that the weak*-limit as t —» 0 of t~l(Tt(x) — x) exists and C(x) is given by this limit. In the applications, however, it is usually difficult to identify X and one looks for appropriated subspaces. By Prop. 2.1, in order to establish the Feller property of T with respect to a given sub-C""-algebra Ao, it suffices to find a subset Ai norm-dense in Ao which is contained in the domain of the w*-generator £ of T'. The Phillips' space X (see Section 3 below) and the domain of £, however, (see [3], [15]) do not need to be a subalgebra of A. In this note we are concerned, in particular, with the class of w*continuous QDS on B(fy) whose generator is associated with quadratic forms

61

£(x) given by (1) where the operators G, Lf satisfy the following hypothesis. (H-min) The operator G is the infinitesimal generator of a strongly continuous contraction semigroup on [), dom(G) is contained in dom(Le), for all i > 1, and, for all u, v € dom(G), we have £(l)[v, u] = 0. These semigroups arise in the study of irreversible evolutions of quantum open systems (see [1], [2], [19], [28]). The above formula (see [12]) is a generalisation of the Gorini, Kossakowski, Sudarshan, Lindblad ([20], [21]) representation of the infinitesimal generator of a norm continuous QDS to unbounded operators G, Lf. It is well-known (see e.g. [12] Sect.3, [14] Sect. 3.3) that, given a domain D contained in dom(G), which is a core for G, it is possible to built up a QDS, called the minimal QDS and denoted 7"(mm)) satisfying the (equivalent) equations:

(v,Tt(x)u) = (v,xu) + I (v,£(Ts(x))u)ds,

(2)

(v,Tt(x)u) = for u,v G D. The minimal QDS can be defined on positive operators x e B(h) as follows: /T-(min) / \

Tf where the maps Tt

/ \ '(x) = sup /r(n) 77 '(x)

are defined recursively by

(3)

for x £ B(h), u,v e D. The equation (2), however, does not necessarily determine a unique semigroup. The minimal QDS is characterised by the following property (see e.g. [14] Th. 3.21): Proposition 2.2. Suppose that (H-min) holds. Then, for each positive x £ 13(1)) and each w* -continuous family (Xt)t>o of positive operators on B(\)) satisfying (2), we have Tt(min)(x) < Xt for all t > 0.

62

Proof. Immediate from the inequality Tt

(x) < Xt for n,t > 0.

D

Let 7^ denote the predual semigroup on the Banach space Xi(f)) of trace-class operators on t) with infinitesimal generator £* . The linear span T> of elements of Ii(f)) of the form \u)(v is contained in the domain of £ » " ' . Thus we can write the equation (2) as follows

This equation shows that the solution to (2) is unique whenever the linear manifold £;mm^ (T>) is big enough. Indeed, the following characterization holds. Proposition 2.3. Under hypothesis (H-min) the following conditions are equivalent: (i) the minimal QDS is Markov (i.e. Tt(min}(I) = 1), (ii) (Tt )t>o is the unique w* -continuous family of positive contractive maps satisfying (2) for all positive x 6 B(fy) and all t > 0, (Hi) the domain T> is a core for £;mm' . We refer to [12] Th. 3.2 or [14] Prop. 3.31 (resp. [14] Th. 3.21) for the proof of the equivalence of (i) and (iii) (resp. (i) and (ii)). Usually (i),...,(iii) are difficult to check. A reasonable and applicable condition implying (i),...,(iii) was given in [9] (see also [14] Sect. 3.5 and 3.6). The following is a useful characterisation of the domain obtained in [17] Lemma 1.1. This is the counterpart at the level of the infinitesimal generator of a well-known property of weak*-continuous semigroups (see [4] Prop. 3.1.23 p. 182): an operator x e B(ty) belongs to the domain of the infinitesimal generator £ of a QDS T if and only if sup t>0 t~l \\Tt(x)— x\\oo < co. Proposition 2.4. Suppose that hypothesis (H-min) holds and that the minimal QDS is Markov. Then the domain of the infinitesimal generator C o/7~(mm) is given by all elements x € #(()) such that the sesquilinear form £(x) on dom(G) x dom(G) (v,u)-^£(x)[v,u] is norm- continuous. We refer to [17] Lemma 1.1 or [14] Prop. 3.33 for the proof.

(4)

63

3.

A' is not an algebra

The Phillips' space of a QMS on B(\j), with generator related to forms (1), often contains compact operators. Proposition 3.1. Suppose that hypothesis (H-min) holds and that the minimal QDS T associated with G, Lf is Markov. Moreover suppose that the operators Le are closed and there exists a domain D*, dense in t), contained in the domain of G* and L\ ((. > \) such that the series ^2f\Lff)(Lff\ converges strongly for all f € D*. Then every compact operator on I) belongs to X. Proof. The strong convergence of the series implies that, for every /i, /2 € D*, the form £(|/i}(/2|) is norm-continuous. Thus, by Prop. 2.4, the operator |/i}(/2| (/i,/2 6 D*) belongs to the domain of C. Therefore, by Prop. 2.1 (b) it also belongs to X. Since finite-rank operators are norm dense in compact operators the conclusion follows. D The Phillips' space, however, is not easily identified (under the hypotheses of the above Proposition fC(tj) might not be Tj-invariant even if G and the Li are bounded as shown in [7]) and, as the domain of the w*-generator £, it might not be an algebra. Counterexamples showing that dom(£) is not an algebra were discussed in [15] on the commutative von Neumann algebra L°°((0, +00); C) by considering the Markov process describing the motion of a point moving towards the origin with unit speed and jumping back in (0, +00) with a probability distribution p when reaching 0. An extension of such semigroups to B(L2((0, +00); C)) provided counterexamples for QMS on a noncommutative von Neumann algebra. W. Arveson ([3]) gave independently the same example with the special choice p(x) = 2aexp(—lax) (exponential distribution). This choice has the advantage of leading to the explicit formula (5) here below. We will show that also the Phillips' space X of the QMS (5) QMS is not an algebra and that the Phillips' algebra (the largest self-adjoint subalgebra of B(\)) contained in X) is not strongly dense. We first introduce the QMS following [3]. Let f) = L 2 ((0, oo); C) and let (Ut)t>o be the semigroup of isometries Utu(s) = u(s — t) for s > t, Utu(s) = 0 for 0 < s < t. Let g be the unit vector in F) g(t) = (Set)1/2 exp(-at) (a > 1) and consider the state u> on B(\)) given by w(z) = ( g , x g ) . Note that Ufg = exp(-at)g and uj(UtxU*) = exp(-2at)w(x) for x £ 6(h) and

64

t > 0. Let (Tt)t>o be the QMS on B(fj) defined by

Tt(x)=u(x)Et + UtxU*

(5)

where Et = l-U t Uj is the projection on the subspace L 2 ((0, t); C) i.e. the multiplication by the indicator function 1 (o t) • Let Et = 1 - Et = U t U*. Since U*E^ = Ut*UtUt* = Uf, for all x e B(fj), we have 7i(a;) - x = (w(i) - x) £t + (C/to:C/t* - x) E(x.

(6)

Therefore, Et and .E^ being orthogonal, we find

\\Tt(x) - x\\ > max {\\(u(x) - *) £t||, | (UtxU; - x) E^\\} .

(7)

Notice that the commutative von Neumann algebra L°°((0,+oo);C) is 7^-invariant and the maps Tt = /7^|i<=o((o,+0o);C) (identifying a function / with the corresponding multiplication operator Mf) define a QMS on L°°((0,+oo);C). Let X = \ f e L°°((0, +00); C) | lim \\Ttf - f\\ = 0 (^ t-»o+ Our first remark concerns this semigroup. Proposition 3.2. A f & L°°((0, +oo);C) belongs to X if and only if f is uniformly continuous on [0, +oo[ and /(O) = w(M/). Proof. By (6) and (7) a function / belongs to X if and only if both - Mf) Et\\,

vanish as t tends to 0+ . The second norm is equal to sup |/0r - *) - f ( x ) \ = sup \f(x + 0 - f ( x ) \ • X>t

T>0

Thus it vanishes as t goes to 0 if and only if / is uniformly continuous. Then the first norm, equal to sup0(M/) — f ( x ) \ , vanishes as t goes to 0 if and only if /(O) = w(M/). D As a corollary we obtain Corollary 3.1. The largest algebra contained in X consists of multiples of the identity operator.

65

Proof. Let /, /* be elements of X. Suppose that also |/|2 belongs to X then w(M]f]I) = |/(0)|2 = |w(M/)| 2 . It follows that /•+00 , (la) I exp(-2ao:) (f(x) - /(O)) 2 dx = 0 Jo i.e. f ( x ) = /(O) for every x > 0.

D

2

We now go back to QMS on B(L ((Q, +00); C)). Proposition 3.2 shows that X is not an algebra. Indeed, if M/ is the multiplication operator by a function / satisfying the conditions of Prop. 3.2, then the square is My2 and condition w(M/2) = /(O) 2 usually fails for the square of /. However any non-constant / satisfying this condition e.g. /(s) = (s + (2a)/(2a + 1)) exp(— s] yields a / 2 which does not satisfy u(Mj) = /(O)2 by the argument of Corollary 3.1. More precisely we can prove the following proposition. Proposition 3.3. Let Ao be a *-algebra contained X. x € AQ, g is an eigenvector for both x and x* .

Then, for every

Proof. Since x, x*, x*x belong to Ao, for all t > 0, \u>(x*x) — u!(x*)u(x)\ = \\(w(x*x) — u>(x*)uj(x)) Et\\ is not greater than ||(w(x*i) - x*x) Et\\ + ||w(x) (u(x*) - x*) Et\\ + ||i* (w(z) - x) Et\\ . Thus u(x*x) = u(x*)u(x) by (7). It follows that w(\x - u(x)\2) = 0 i.e. In a similar way, exchanging x and x*, we find x*g = u>(x*)g.

D

Note that the conclusion of Prop. 3.3 also holds for a *-algebra contained in the domain of £ ([3] Prop, p.9, [15] Prop. 3.3 p.85). It could also be proved by the same arguments of Prop. 9 in [3], or deduced immediately from this result and Prop. 2.1, that the strong closure of the largest self-adjoint algebra contained in X consists of all operators x e B(t)) such that both x and x* have g as an eigenvector. 4. The JC(l))-Feller property We begin by the following characterisation of .AC (I)) -Feller QDS on B(t)) proved in [7] Prop. 2.1. Proposition 4.1. Let T be a QDS on S(f)). The following conditions are equivalent: (1) T is K,(\))-Feller,

66

(2) /C(h) is Tt invariant for all t > 0. Proof. By the w*-continuity of T, for every x & /C(h) and a 6 2"i(/i), the map t —* tr(
CQ(x) = £(x]

generates a strongly continuous contraction semigroup on /C(F)). Proof.(l) => (2). Each compact operator x belongs to the domain of the generator of the strongly continuous semigroup (Tt^^j)t>o if and only if limt-^o^1 (T~t(x) — x) exists in the strong or, equivalently (e.g. by [4] Cor.3.1.8 p.168)), weak topology. However, since the dual space of JC(h) is Zi(f)), the above limit exists in the weak topology if and only if it exists in the weak* topology i.e. if and only if x edom(£). Therefore the operator £0 is the strong generator of CZt|jc(fj))t>o(2) =>• (1). Condition (2) implies that, for all positive A big enough, the operator (A — £) : dom(£) n /C(f)) —> /C(h) is one-to-one. Therefore /C([)) is invariant under the resolvent operators (A — £}~l. Then, from the well-known formula Tt(x)= lim v((n/t)(n/t-Crl}n(x-)= n—>oo

lim ((n/t}(n/t - A))"1)" W

n—>oo

(x € /C(h)), it follows that /C(h) is Trinvariant. Then 1 is £(h)-Feller by Prop. 4.1. D The above theorem clearly generalises our characterisation (Th. 3.2 [7]) of ^C(t))-Feller norm-continuous QDS. Indeed, for bounded G and Li with ~Y^i LfL/e strongly convergent, condition (2) holds if and only if the operator Y^t \LfU)(LfU\ is compact for each u £ h.

67

We do not know a simple characterisation of £(h)-Feller QMS associated with operators G, LI depending only on the G, L^. However we have the following Theorem 4.2. Suppose that hypothesis (H-min) holds and: (i) the operators Lt are closed, (ii) there exists a domain D* dense in h, contained in dom(G*} and in dom(Lf) (t > 1) which is a core for G* such that, for each f £ D* , we have (8)

where b is a positive constant independent of f . Then 1i(h) is Tt -invariant for each t > 0 and ||7^(a;)||i < 2exp(&£)||o;||i for each x € TI(/I). In particular T is K,(fy)- Feller. A proof was given in [7] Th. 4.1 and 4.2. Here we outline another one inspired by [12] Sect. 2. We first introduce some notations. Let V be the linear manifold in B(tj) generated by rank-one operators |u 1 )(u 2 | with ui,«2 € D* and let Vs.a. be the submanifold of self-adjoint elements of V. Clearly the semigroup T^ defined by (3), can be also viewed as a semigroup on the Banach space Is,a. of self-adjoint, trace class operators. Then it defines a strongly continuous contraction semigroup on Is.a., Vs.a. is a core for its generator Q, and

G(x) = G*x + xG for x € Vs.a.- This follows with minor modifications (taking D* instead of dom(<7*)) from [12] Lemma 2.1 p. 170. Let J be the linear map

J : Vs.a. -> Is.a. ,

J(x) =

This is well defined under hypotheses (i) and (ii) of Th. 4.2. Lemma 4.1. Under the hypotheses of Th. 4-%f°r all X > b and all x G Vs.a. we have

68

Proof. Any x 6 Vs.a. can be represented as a finite sum '^lmxm\em){em\ with em € D*, (em)m orthonormal, xm € K. Clearly ||o;||i = £]m kml and J(\- gy1 W = E*™ E j/

e>i °

e A

" * I W fi ™> (L\P*tem\dt.

For all m, by the inequality (8), integrating by parts, we have "ejfit

< 1 - < * - » > Jof Then the stated inequality follows.

\2dt. D

Proof (of Th. 4.2). By a well-known perturbation result in semigroup theory (see e.g. [24] Th. 3.2 p.81), for all r e]0,1[, the closure of the operator Q+rJ, defined on Vs.a. generates a strongly continuous semigroup (<S( )t>o on Is.a. satisfying the equation (2) with rLf replacing Le and ||^tr)(a;)||i < exp(6t)||a;||i. Note that, for all positive a 6 Ji(t)), the map r —> <S( (cr) is increasing. Therefore, letting r go to 1, we can define a strongly continuous semigroup (S\ )t>o on 1s.a, satisfying (2). Then, clearly, <St(1)(o") > Tt(min}(cr) for all positive a € Ji(f)) by Prop. 2.2. Moreover, it could be shown by an approximation argument (constructing the maps <St(r'n) like Tt(n\ showing that 5t(r'") < Tt(n} and letting n go to infinity) that <St (cr) < Tt (cr). Hence S^ coincides on Is.a. with the minimal QDS. It follows that Ji(f)) is 7^-invariant and Tt satisfies the claimed inequality. The 7^-invariance of compact operators follows then from the normD density of Ji([)) in K.(t)). 5. The C*-algebra M Let f) = L 2 (R d ;C) with position and momentum operators defined in the Introduction. We would like to establish the Feller property of some QMS with respect to a reasonable extension M. of /C([j). Theorem 5.1. The C*-algebra M. generated by the identity 1 and the operators f ( p ) , g(q) with /,g € C°(R d ;C) coincides with the C*-algebra of operators in B(t)) of the form zl + f(p) + g ( q ) + K

zeC

,£CRd;C

and K

(9)

69

The proof will be divided into several lemmas. Lemma 5.1. For f, g € Co(M d ;C) the operator f(p)g(q) is compact. Proof. Indeed, when / and g, moreover, have compact support, f(p)g(q) is a Hilbert-Schmidt operator by [27] Th. XI.20 p.47. If f,g e C§(Rd;C) then, let (/„)„>!, (#„)„> i be sequences of continuous functions with compact support converging uniformly to / and g respectively. For all n,m > 1 the norm \\fn(p)gn(q) - fm(p)gm(q)\\ can be estimated by

\\fn(p) (gn(q) - gm(q)) || + 1| (fn(p) -

fm(P))gm(q)\\

|/n - fm\\ + \\gn - gm\\)

It follows then that f(p)g(q) is a compact operator since it is the limit in norm of fn(p)gn(q). D The above Lemma shows, in particular, that the linear manifold of elements of B(fy) represented in the form (9) is a *-algebra. The next lemma is inspired by [8] Section 2. Let us recall first that the Fourier transform is the unitary operator U on f) defined by (Uu)(x) = u(x) = (27r)-d/2

JR*

ei(x'y}u(y)dy

with inverse (U*u)(x] = (27r)- d / 2 /

e-i{x'y}u(y)dy.

JTSLd

Lemma 5.2. Let 5 G f) and let fe (s > 0) be the Gaussian density with mean 0 and variance £

The operator fe(p)g(q) is a Hilbert-Schmidt operator on f) with kernel

Proof. The operator fe(p)g(q) is a Hilbert-Schmidt operator by [27] Th. XI.20 p.47. We find now its kernel. By the well-known properties of the Fourier transform of fs(p) coincides with the operator UM/e U* where Mfe denotes the multiplication operator by the function f£. Clearly, g(q) is the multiplication operator Mg.

70

For each continuous function u on Rd with compact support the function (UMfcU*Mg)u is given by f e-i
J^fc(z}dz

JEd

= (^rd/2 I dy I (27r)-d/2 / d Jut* \ jR = (27r)-d/2

j(*-y^fe(z}dz}g(y}u(y) J

fe(x-y)g(y)u(y}dy.

Therefore we have ((UMf.U*Ma)u)(x)=

[ k(x,y)u(y)dy.

(10)

JHLd

Since continuous function with compact support are dense in F) , the identity \ k ( x , y ) \ 2 d x d y = (27r)-d||7e||2 • \\g\\2 = (27r)-d||M|2 • ||5||2 < oo (|| • || denotes the norm in h) shows that (10) holds for all u G f) and the Lemma is proved. D Lemma 5.3. Let g £ I) and /£ be as in Lemma 5.2. The operator K = 9((l)fe(p)g((l) is a Hilbert- Schmidt operator with kernel k(x, y) = (27r)- d / 2 g(x)/ £ (x - y)g(y). Proof. For all n > 1 let gn(x) = max{— n, min{g(o;), n}}. Clearly gn(q) is a bounded operator and, by Lemma 5.2, gn(q_}fe(p)g(q) is a compact operator Kn with kernel kn(x,y) = (2Tr)~d/2gn(x)fs(x — y)g(y). The conclusion follows from the inequality \k(x,y)-kn(x,y)\'2dxdy and the convergence of gn to g in h.

D

Lemma 5.4. For all g G f) we have || - || - linjig(q)fe(p)g(q)

= (2^-d

Proof. Since f£(x-y) = (27r)- d / 2 e- e l :c - s 'l 2 / 2 , the operator g(q)f£(p)g(q) (27r)-d/2|5)(5| has kernel ( x , y ) -> (27r)- d / 2 5 (x) e-e\*-'2 - l

g(y).

-

71

The conclusion follows since, by dominated convergence, the above function goes to 0 in L2(E.d x R d ; C) as s goes to 0. D Lemma 5.5. Every compact operator on f) belongs to A4. Proof. For each u £ (} let (gn)n>i be a sequence of continuous, squareintegrable functions converging to u in t). By Lemma 5.4, the onedimensional projection \gn)(gn\ belongs to M. Moreover, \gn}(gn converges in norm to |u)(u|. Thus also \u)(u belongs to M. Now it is easy to see that every finite rank operator belongs to M. so that every compact operator also belongs to M. D We wish to thank H. Comman who pointed out that the following Lemma is an immediate consequence of Weyl's theorem. Lemma 5.6. Let /, g 6 Co(R d ;C) and let K be a compact operator on f). The following equalities hold \\f(p) + K\\= |/(p)[|,

\\g(q) + K\\ = \\g(q)\\.

Proof. We prove the first identity. Since K is compact, by the Weyl's theorem, f(p) + K and f(p) have the same essential spectrum. Since all the spectrum of f(p) is essential, then f(p) + K and f(p) have the same spectrum. Hence they have the same norm. The proof of \\g(q) + K\\ = \\g(q)\ is identical. D Lemma 5.7. Let /, g £ Cg(K d ;C) and let K be a compact operator on f). The following inequality holds \\f(p)+g(q)+K\\>m^{\\f(p)\\,\\g(q)\\}. Proof. Let r0 e Rd such that |<7(r 0 )| = ||#|| and let tp be the function

Clearly \\g(q)\\ = \\((q)\\ = 1- By Lemma 5.1, the operator -^ = (P(
+ K\\ = ||^(g)|| - ||/(p) + g(q) + K\\

>\\(V9)(q)\\=\\g(q)\\.

Therefore, applying the unitary Fourier transform U, we have also + 9(q) + K\\ = \\U*(f(p) + g(q) + K)U\\

72

This proves the Lemma.

D

Lemma 5.8. Let f , g £ C$(R.d;C), z e C and let K be a compact operator on I). The following inequality holds

Proof. Let r e Rd and let <pr : Rd -> C be the function (pr(x) = (l + |x — r| 2 ) . The above arguments and results lead to the inequalities f(p) + g(q) + K|| = ||w(g)|| • ||zl + f(p) + g(q)

Since the function tprg converges uniformly to 0 as r tends to infinity, we have that limr_oo ||(vrfl)(9)|| = 0, and the conclusion follows taking the limit in the previous relation. D We now prove the main result of this section. Proof, (of Th. 5. 1) The set of operators in B(h) of the form (9) is an algebra by Lemma 5.1 and it is contained in M. by Lemma 5.5. In order to conclude, we need to show that it is complete. Let us consider a Cauchy sequence (zn\ + f n (p) + g n (q) + K n ) n >i- Lemma 5.8 implies that the sequences (zn)n>i, and, hence, (fn(p) + gn(q) + Kn)n>i are Cauchy in the complex numbers and in B(fj) respectively. Then Lemma 5.7 shows that (/«)n>i, (ff n )n>i and, hence, (Kn)n>i are Cauchy in Cg(R d ;C), Cg(R d ;C), and in H(f)) respectively. The conclusion follows then from the completeness of the above Banach spaces. D Remark. Notice that the previous lemmas also show the uniqueness of the representation of every element of M. in the form (9). Remark. It is worth noticing here that the inequality g(q) + K\\> \\K\\

(11)

does not hold as shows the following example. Let f) = L 2 (R; C) and let e be the element of f) defined by e(x) = ^-(l-\x\)l/\

a: €[-1,1]

73

and e(x) = 0 elsewhere. Clearly \\e\\ = 1. Let g(x) — |e(o:)|2 and K = —\e}(e\. We show now that (11) does not hold. To this end we estimate the norm of g(q) + K. For all u € f) with ||u|| = 1 we have (x)|2|u(x)|2do; <

f

V/R

e(x)u(x)dx

\e(x)\2\u(x)\2dx

Taking the unit vectors (u£; 0 < e < 1) defined by u£(x) = (3/2)1/V/2 (s - M) (l [0 , £] (x) - l[_ £ , 0 [0r)) , it is easy to see that the above upper bound is sharp. Moreover, by the Schwarz inequality, we have (U,(\e(q)\2- e)(e\}u}> f |e(x)| 2 | U (x)| 2 di-2 /' \e(x)\*\u(x)\*dx J-\ J-i =- / J-i

\e(x)\2\u(x)\2dx

> -1 f

\u(x)\*dx > ~.

It follows that IIH<7)| 2 -|e)< e |||=3/4.

6. Applications Our first application will be to the quantum Ornstein-Uhlenbeck semigroup introduced in Section 1. Th. 5.1 in [10] shows that the quantum OrnsteinUhlenbeck semigroup is /C(f))-Feller. This follows also as an application of Th. 4.2 here. Moreover, Proposition 6.1. The quantum Ornstein-Uhlenbeck semigroup is MFeller. Proof. This is obvious since it is identity preserving, /C(h)-Feller and its restriction to the commutative C*-algebra of continuous functions vanishing at infinity of position or momentum operators yields the classical (strongly continuous) Ornstein-Uhlenbeck semigroup (see e.g. [16] Sect.3). D

74

The algebra A4, however, is too big for several similar QMS. Indeed, consider the QMS given by Tt(x) = e^ltNxeltN where N = '}^ia*iat is the number operator or, in other words, the classical Ornstein-Uhlenbeck operator. The arguments of Section 5, together with the analytical extension of Mehler's formula lead to the inequality

\\e-itNg(q)eitN-g(q)\\>\\g(q)\\ for all non-zero g & Co(K rf ;C) and all t in a right neighborhood of 0. It follows that T is not .A/f-Feller. As a second application we show that a QMS arising in Quantum Optics is /C(h)-Feller. This could be proved for a lot of QMS in the physical literature ([2], [11], [19], [23], [28]) as a straightforward application of Th. 4.2 but, for lack of space, we will discuss only the model for absorption and stimulated emission from [19]. We first introduce the QMS following [18]. Let I) = / 2 (N) and let T be the minimal QDS associated with Li = z/a*a, L2 =pa, H = £(a + a*), G = -2"1 (LJLi + L^La) - iH where fj,, v > 0, £ e R and a and a* are the usual annihilation and creation operators on f). It has been shown in [18] that T is Markov i.e. hypothesis (H-min) holds. Proposition 6.2. The absorption and stimulated emission QMS is JC(J))Feller. Proof. Clearly the operators LI, 1/2 are closed. Taking as D* the domain of the number operator TV = a*a, a straightforward computation shows that the inequality (8) holds for every / 6 D* with b = /A The conclusion follows from Th. 4.2. D References 1. Accardi, L.; Lu Y.G.; Volovich, I.V.: Quantum Theory and its Stochastic Limit. Springer Verlag (2001) to appear. 2. Alicki, R.; Lendi, K.: Quantum Dynamical Semigroups and Applications. Lecture Notes in Physics, 286. Springer (1987). 3. Arveson, W.: The domain algebra of a CP-semigroup. Preprint, 2000. http://www.math.berkeley.edu/- arveson/texflies.html 4. Bratteli, O.; Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics I. Springer-Verlag, 1979. 5. Bratteli, O.; Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics II. Springer-Verlag, 1981.

75

6. Butzer, P.L. and Berens, H.: Semi-groups of operators and approximation. Die Grundlehren der mathematischen Wissenschaften, Band 145 SpringerVerlag New York Inc., New York 1967. 7. Carbone, R.; Fagnola, F.: The Feller property of a class of Quantum Markov Semigroups. To appear in: Proceedings of the VI Simposio de Probabilidad y Procesos Estocasticos. Guanajuato (Mexico) 23-27 May 2000. 8. Cassinelli, G.; Varadarajan, V.S.: Some remarks on a problem of Accardi. To appear in: Infin. Dimens. Anal. Quantum Probab. Relat. Top. 2001. 9. Chebotarev, A.M.; Fagnola, F: Sufficient conditions for conservativity of quantum dynamical semigroups. Preprint n.308. Genova, May 1996. J. Funct. Anal. 153, n. 2, p. 382-404 (1998). 10. Cipriani, F.; Fagnola, F.; Lindsay, J.M.: Spectral Analysis and Feller Property for Quantum Ornstein-Uhlenbeck Semigroups. Comm. Math. Phys. 210 (2000) 1, 85-105. 11. D'Ariano, G.M.; Sacchi, M.F.: Equivalence between squeezed-state and twinbeam communication channels. Modern Phys. Lett. B 11 (1997) 29, 12631275. 12. Davies, E.B.: Quantum dynamical semigroups and the neutron diffusion equation. Rep. Math. Phys. 11 (1977), no. 2, 169-188. 13. Chung, K.L.: Doubly-Feller processes with multiplicative functional. Seminar on stochastic processes, 1985 Gainesville, 1985), 63-78, Progr. Probab. Statist., 12, Birkhauser Boston, Boston MA, 1986. 14. Fagnola, F.: Quantum Markov Semigroups and Quantum Markov Flows. Proyecciones 18, n.3 (1999) 1-144. 15. Fagnola, F.: A simple singular quantum Markov semigroup. Proceedings of the Third International Workshop Stochastic Analysis and Mathematical Physics ANESTOC '98 Birkhauser 2000 p. 73-88. 16. Fagnola, F.; Rebolledo, R.: An ergodic theorem in quantum optics. In C. Cecchini (ed.): "Contributions in Probability". Forum, Udine 1996. 17. Fagnola, F.; Rebolledo, R.: The approach to equilibrium of a class of quantum dynamical semigroups. Infinite Dimensional Analysis, Quantum Probability and Related Topics. 1 n. 4 (1998), 561-572. 18. Fagnola, F.; Rebolledo, R.: Lectures on the Qualitative Analysis of Quantum Markov Semigroups. CIRM-Volterra International School "Quantum Interacting Particle Systems" Levico Terme, September 200. Preprint. Santiago, April 2001. 19. Gisin, N.; Percival, I.C.: The quantum-state diffusion model applied to open systems. J. Phys. A: Math. Gen. 25 (1992) 5677-5691. 20. Gorini, V.; Kossakowski, A.; Sudarshan, E.C.G.: Completely positive dynamical semigroups of AT-level systems. J. Math. Phys. 17 (1976), no. 5, 821-825. 21. Lindblad, G.: On the generators of quantum dynamical semigroups. Comm. Math. Phys. 48 (1976), no. 2, 119-130. 22. M.Ohya, D.Petz: Quantum Entropy and its Use, Springer 1995. 23. Olkiewicz, R.: Structure Algebra of Effective Observables in Quantum Mechanics. Ann. Phys. 285 (2000).

76

24. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, 1975. 25. Petz, D.: An Invitation to the Algebra of Canonical Commutation Relations. Leuven University Press 1990. 26. Phillips, R.S.: The adjoint semigroup. Pacific J. Math. 5, 269-283 (1955). 27. Reed, M.; Simon, B.: Methods of modern mathematical physics. III. Scattering theory. Academic Press. New York-London, 1979. 28. Schack, R.; Brun, T.A.; Percival, I.C.: Quantum-state diffusion with a moving basis: Computing quantum-optical spectra. Phys. Rev. A 55 n.4, 26942697. 29. Stinespring, W.F.: Positive functions on (7*-algebras, Proc. Am. Math. Soc., 6 (1955), 211-216.

ASYMPTOTIC BEHAVIOUR OF MARKOV SEMIGROUPS ON NON COMMUTATIVE /^-SPACES *

EDUARD YU. EMEL'YANOV Sobolev Institute of Mathematics, Acad. Koptyug pr. 4, 630090 Novosibirsk, Russia E-mail: [email protected] Current address: Mathematisches Institut, Universitdt Tubingen, Auf der Morgenstelle 10, D-72076 Tubingen, Germany E-mail: emelyanov@uni-tuebingen. de

MANFRED P. H. WOLFF Mathematisches Institut, Universitdt Tubingen, Auf der Morgenstelle 10, D-72076 Tubingen, Germany E-mail: manfred. wolff@uni-tuebingen. de

1. Introduction During the last two decades several important results about the asymptotic behaviour of Markov semigroups on L1(O, E,/i) with a-finite measure /j, were established (see [Ko] or [LM] for a survey and applications). We will generalize some of them to the noncommutative setting. For simplicity, in this paper we deal with Markov operators only, but with exception of Theorem 2.1, all results hold also for power-bounded positive operators. Another more general and more sophisticated approach to such type of problems is presented in [EW2], where in particular the results for general positive semigroups are proved. *2000 mathematics subject classification, primary 47a35, secondary: 47cl5, 47b60, 47b65, key words and phrases, markov semigroup, constrictive semigroup, quasi order bounded attractor, statistical stability.

77

78

It is known that every space Ll(Cl, S,/i) is the predualof Z/°°(rj, £,/u) whenever the measure space (fi, £,/z) is cr-finite. On the other hand, it is also known that every non commutative L1-space is the predual of some W*algebra. So in the following we consider a fixed W*-algebra M. together with its unique predual ,M» and dual M*. For x < y in (A / t*) so , denote by [x,y] the order interval [z € (M*)Sa. '• x < z < j/}, and by B ( M f ) s a the closed unit ball of (A / f») S a- A subset A of (M.*)Sa is called quasi order bounded if there exist 0 < z e M* and 0 < 77 < 1 such that

We will denote the set {/ e (M*)+ : \\f\\ = 1} which is usually called set of all normal states on M by S(M*). Recall that a positive operator T on A4* is called Markov operator whenever the unit II of M. is a fixed point of its adjoint T'. Notice that for a Markov operator T the relations ||T|| = 1 and T(S(MJ) C S(M.) hold. Let T = (Tt)teJ be a one parameter semigroup of Markov operators on M*, where J = N U {0} or J = M + . A set A C (M*)aa will be called a real attractor of T if lim

t —>oo

for any x G -B(x,).,a- ^e W1^ denote the set of all real attractors of T by Attrr(T). Following to Lasota, Li, and Yorke [LLY] we call T constrictive whenever K e Attrr(T) for some compact K. For a constrictive T there exists (see [LLY] for the commutative L1 and [Vu], [LLY] for a general Banach space setting) a decomposition M, := M° ® M:

(*)

into T-invariant subspaces M°, Ml such that M° = {x e .A/U : lim ||Tta;|| = 0} n —KX>

and

dim(M;) < oo.

So, asymptotically a constrictive T can be investigated by linear algebra methods only, and henceforth it is clear that it is important to find conditions under which T will be constrictive. The following important result in this direction was stated by Komornik and Lasota [KL] : Theorem 1.1. [KL] T be a Markov operator on the space Ll(£l,A,n), where (JL is a-finite. Then T — (Tn)^L1 is constrictive if and only if T possesses a quasi order bounded real attractor.

79

Recall that a V7*-algebra M. is called atomic if every non zero projection majorizes a non zero minimal projection. For example the algebra M. = B(H] of all linear bounded operators on a Hilbert space H is atomic. It will be shown that the assertion of Theorem 1.1 holds also for the predual of an atomic M/*-algebra (see Theorem 2.2 bellow). We say that a Markov semigroup T on M* is statistically stable whenever there exists a T-fixed state u £ <S(.A/U) such that lim Ttf = H(/)u t —>oc

for each / G M.*. Analogously to the commutative case 0 ^ h € (A'(*)+ is called a nontrivial lower-bound element for T if lim \\(h — Tt/)+|| = 0 t —>oo

holds for all 0 < / e .M,, ||/|| = 1. It is well known and due to Lasota [La], Theorem 1.1 (see also [LM], Theorems 5.6.2 and 7.4.1) that a one parameter Markov semigroup on L1^, S, p.) is statistically stable if and only if it has a nontrivial lower-bound element. The generalization of this result to the non commutative setting is due to Ayupov and Sarymsakov and it was announced without proof in [AS], Theorem 2.4. Below we give a slightly more general result (see Corollary 2.1) with a very short proof. 2. Main results We begin with the following proposition. Proposition 2.1. Let M.* be the predual of the W*-algebra M., and let T = (Tn)nejy be a discrete semigroup of Markov operators on M* which possesses a quasi order bounded real attractor. Then T is mean ergodic. Proof: Let [—y, y] + rjB^M.),* be an attractor of T, where 0 < 77 < 1. By Sine's mean ergodic theorem ([Kr], p. 74) and a simple linearity argument it is enough to show that for every fixed point if} G M.sa of T' there exists a fixed point w 6 (M*)+ of T which satisfies (ijj, w) ^= 0. Let M.sa 3 tp ^ 0 be a fixed point of T'. We may assume that \\i(>+\\ = = 1. Set e := (1 — ?y)/3 and take some x e (-M*)sa which satisfy = 1 and (V>+,:r) > 1 — e. We have that x = x = 1 and

Thus {V<, x\) = (2V>+, \x\) - (|Vi |x|> > 2(1 - e) - 1 = 1 - 2e. Set An(T) = ^ ^fcZ0 T fc . Let x" £ .M* be a u)*-cluster point of {An(T)|x|}neN. Then (Tn)"x" = x". Since lim dist(T" x\, [~y,y] +

80

r)BM,) = 0 and [—y,y] is weakly compact in M* by a result of Akemann [Ak], Theorem 11.2(2), we obtain x" & {v e M* : -y < v < y} + rjBM" C M, + nBM* • Take the positive projection R : M* —> .M* according to [Sa],Prop. 1.17.7. Then (!M- ~ R)x" € r?Bx«, and hence

= {x", V>+) - ((IM. - R)x", ,

> (x", V) - TI = (V>, N) - 7 ? > l - 2 £ - ? 7 = £ > 0 . Moreover TmAc" = TmR(Tn)"x" > TmR(Tn)"Rx" = TmRTnRx" = Tm+nRx" > 0. Thus (TnRx"}n is decreasing in (.M,)+. So, w := lim T"fix" exists. n—>oo

Clearly Tw = w, and (-0, w) = (y>, fix") > 0. o We apply this proposition to obtain an easy proof of the following assertion. Corollary 2.1. ([AS], Theorem 2.4) Let T be a (not necessarily continuous in case of J = ]R+) Markov semigroup on M.*. Then the following assertions are equivalent: (i) T is statistically stable. (ii) There is 0 ^ h £ (M»)+ such that for every f e S(M*), and for every t e J there exists ft e (M*)+ with lim ||/t|| = 0, and Ttf + ft > h for all t—>00

t 6 J.

(iii) T has a nontrivial lower-bound element. Proof: (i) => (Hi) : Let 0 ^ u e (X*)+ satisfies lim T(/ = fl(/)u for each £ —>cx) / € A't*. Obviously, u is a nontrivial lower-bound element for T. (m) =^> (ii) : Let 0 < h 6 .M* be a nontrivial lower-bound element for T. Then for any / 6 <S(.M*) the condition (ii) is satisfied with ft := (Ttf — h)for all t 6 J. (ii) => (f) : In the discrete case T = (T")^L0 easy computations show that *)sa : lim ||T"0|| = 0} =

81

Moreover by Lemma 2.1, there exists a T-invariant normal state u. Therefore, we obtain (M*)sa = (M*)°a © TR.u, the decomposition into two Tinvariant closed subspaces, and so

lim Tnf = fl(/)u (V/ 6 M»).

n—>oo

In the case of a not necessarily continuous Markov semigroup T = (^~*)telR , one can apply what was proved for the discrete case and repeat the short and elegant arguments due to Lasota and Mackey [LM] , Proof of Theorem 7.4.1, which works similarly also for noncommutative L1-spaces. o Remark: In [AS], Theorem 2.4, only the equivalence (i) o (if) was stated for discrete semigroup of Markov operators. However Corollary 2.1 and Theorem 2.4 in [AS] are essentially the same. For proving Theorem 2.2 we need the following result ([EW2], Theorem 8) which is stronger then Proposition 2.1. Unfortunately, the proof of this theorem is too long to be reproduced here. Theorem 2.1. [EW2] Let T = (Tt)te j be a discrete or strongly continuous semigroup of positive operators on the predual M.* of the W* -algebra M.. IfT possesses a real attractor [— y , y ] + tyB^.)^ for some 0 < r\ < 1 and T

m-l

some y > 0 then w := lim m — J3 Tky (respectively, w := lim - fTtydt) m-too

k_Q

T->oo

Q

exists and moreover j^— [— w,w] is a real attractor ofT. In particular, under the same assumptions as in Proposition 2.1, the semigroup (Tn)^L1 has an order bounded real attractor, and consequently the operator T is weakly almost periodic. Now we are in a position to state the main result of this paper which is given here in a form slightly different from the one in [EW2], Corollary 15, where it was stated for completely positive operators only. Theorem 2.2. Let T := (T t ) t6 j be a discrete or strongly continuous semigroup of Markov operators on the predual M.* of an atomic W* -algebra M. Then the following conditions are equivalent: (i)

T is constructive.

(ii) T possesses a quasi order bounded real attractor.

82

Proof: (i) =*> (n) : It is easy to see that T possesses a quasi order bounded real attractor, since for every compact set K C (A4*)sa and £ > 0 there exist z € (.M*)+ such that K C [—z, z] + e.B(.M.)Ja(n) => (i) : Let T := T\. Then (Tn)™=0 possesses the same quasi order bounded real attractors as T. The operator T is weakly almost periodic by Theorem 2.1. Moreover, each order interval in the predual of an atomic W-algebra is compact as follows by combining Theorem 11.2(2) in[Ak] and Corollary III.5.11 of [Ta]. Finally, applying [EW1], Corollary 8, we obtain that there exist a compact K £ Attrr((Tn)%LQ). This finishes the proof in the case when J = NQ. Now if (T t ) t >o is a strongly continuous semigroup, it is enough to note now that ^[0,1] := {Ttx : x £ K, t e [0,1]} is a compact real attractor of T. o We give an application of Theorem 2.2. Recall that the semigroup T of positive operators on the predual M.* of the W* -algebra M. is called irreducible if its adjoint semigroup T on M. does not possess o-(M, M.*}closed invariant hereditary subcones other than {0} or M.+ (see [Gr2], p. 388). An operator T is called asymptotically periodic whenever there exists a periodic operator Q (this means Qp+1 — Q for some p e N) of finite rank such that the sequence (T™ — Qn)^L0 converges in the strong operator topology to 0. Corollary 2.2. ([EW2], Corollary 16) Let T be a completely positive Markov operator on the predual M.* of the atomic W*-algebra M. and assume that T := (T")^=0 is irreducible and possesses a quasi order bounded attractor. Then T is asymptotically periodic. Proof: Theorem 2.2 implies that T is constrictive, so dim(MJ) < oo. By [Grl] the peripheral point spectrum cr(T) n {A e (D : |A| = 1} is a finite group. Hence T is periodic on Ml where we use the notation given in the decomposition (*). o

ACKNOWLEDGEMENT The first author is grateful for generous support of the Alexander von Humboldt Foundation during his stay at the University of Tubingen.

83

References Ak.

C.A. Akemann, The dual space of an operator algebra, Trans. Arner. Math. Soc. 126 (1967), 286-302. MR 34#6549 AS. Sh.A. Ayupov and T.A. Sarymsakov, Markov operators on quantum probability spaces, Probability theory and applications, Proc. World Congr. Bernoulli Soc, Tashkent/USSR 1986, Vol. 1 (1987), 445-454. EW1. E.Yu. Emel'yanov and M.P.H. Wolff, Quasi constricted linear operators on Banach spaces, Studia Math. 144 (2001), no.2, 169-179. EW2. E.Yu. Emel'yanov and M.P.H. Wolff, Asymptotic properties of positive operators on preduals of W*-algebras, (submitted 2001). Grl. U. Groh, The peripheral point spectrum of Schwarz operators on C* algebras, Math. Z. 176 (1981), 311-318. MR 82i:46088 Gr2. U. Groh, Spectral theory of positive semigroups on W*-algebras and their preduals, in R. Nagel (ed.) One-parameter semigroups of positive operators, Lecture Notes in Math. 1184, Springer (1986). MR 88i:47022 Ko. J. Komornik, Asymptotic periodicity of Markov and related operators, Dynamics reported, Dynam. Report. Expositions Dynam. Systems (N.S.) 2 (1993) 31-68, Springer, Berlin. MR 961:47017 KL. J. Komornik and A. Lasota, Asymptotic decomposition of Markov operators, Bull. Polish Acad. Sci. Math. 35, no.5-6 (1987), 321-327. MR 88m:47014 Kr. U. Krengel, Ergodic Theorems, De Gruyter, Berlin - New York (1985). MR 871:28001 La. A. Lasota, Statistical stability of deterministic systems, Equadiff 82 (Wurzburg, 1982), 386-419, Lecture Notes in Math, 1017, Springer,. Berlin, (1983). MR 85d:28012 LLY. A. Lasota, T.Y. Li, and J.A. Yorke, Asymptotic periodicity of the iterates of Markov operators, Trans. Amer. Math. Soc. 286 (1984), 751-764. MR 86m:47010 LM. A. Lasota and M.C. Mackey, Chaos, fractals, and noise. Stochastic aspects of dynamics, Second edition. Applied Mathematical Sciences, 97. Springer-Verlag, New York, (1994). xiv+472 pp. MR 94j:58102 Pe. G.K. Pedersen, C*-algebras and their automorphism groups, Academic Press London New York San Francisco (1979). MR 81e:46037 Sa. S. Sakai, C*-algebras and W*-algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 60. Springer-Verlag, New York - Heidelberg (1971). MR 56#1082 Si. R. Sine, Constricted systems, Rocky Mountain J. Math. 21 (1991), 13731383. MR 93a:47006 Ta. M. Takesaki, Theory of Operator Algebras I, Springer-Verlag, New York - Heidelberg, (1979). MR 81e:46038 Vu. K.V. Vu, Asymptotic almost periodicity and compactifying representations of semigroups, Ukrain. Mat. Zh. 38 (1986), 688-692. MR 88j:22003

RECOGNITION AND TELEPORTATION

KARL-HEINZ FICHTNER Friedrich-Schiller-Unversitat Jena, Fakultat jur Mathematik und Informatik, Institut fur Angewandte Mathematik, 07740 Jena, Germany, E- Mail: fichtner@mathematik. uni-jena. de

WOLFGANG FREUDENBERG Brandenburgische Technische Universitat Cottbus, Fakultat 1, Institut fur Mathematik, PF 101344, 03013 Cottbus, Germany, E-Mail: freudenberg@math. tu-cottbus. de MASANORI OHYA Department of Information Science, Science University of Tokyo, Noda City, Chiba 278-8510, Japan, E-Mail: [email protected]

1. Introduction We study a possible function of brain, in particular, we try to describe several aspects of the process of recognition. In order to understand the fundamental parts of the recognition process, the quantum teleportation scheme3'2'6'7 seems to be useful. We consider a channel expression of the teleportation process that serves for a simplified description of the recognition process in brain. It is the processing speed that we take as a particular character of the brain, so that the high speed of processing in the brain is here supposed to come from the coherent effects of substances in the brain like quantum computer, as was pointed out by Penrose. Having this in our mind, we propose a model of brain describing its function as follows: The brain system BS =X is supposed to be described by a triple ( B(7i), S(H), A*(G) ) on a certain Hilbert space Ti where 5(W)is the set of all bounded operators on H, <5(W) is the set of all density operators and A*(G) is a channel giving a state change with a group G. Further we assume the following: 85

86

(1) BS is described by a quantum state and the brain itself is divided into several parts, each of which corresponds to a Hilbert space so that Ji =@kHk and = ®kfk, f k £ &(Hk)- However, in this paper we simply assume that the brain is in one Hilbert space T~L because we only consider the basic mechanism of recognition. (2) The function (action) of the brain is described by a channel A*=®fcA£. Here as in (1) we take only one channel A*. (3) BS is composed of two parts; information processing part "P" and others "O" (consciousness, memory, recognition) so that X =Xp ® XQ, T~L Thus in our model the whole brain may be considered as a parallel quantum computer9 , but we here explain the function of the brain as a quantum computer, more precisely, a quantum communication process with entanglements like in a quantum teleportation process. We will explain the mathematical structure of our model. Let s = {s1 , s2, • • • , sn} be a given (input) signal (perception) and s = {s 1 ,^ 2 , • • • , s™} the output signal. After the signal s enters the brain, each element s j of s is coded into a proper quantum state p> G S (Hp) , so that the state corresponding to the signal s is p = ®jpj. This state may be regarded as a state processed by the brain and it is coupled to a state po stored as a memory (pre-conciousness) in brain. The processing in the brain is expressed by a properly chosen quantum channel A* (or Ap<8> A£>). The channel is determined by the form of the network of neurons and some other biochemical actions, and its function is like a (quantum) gate in quantum computer8'10. The outcome state ~p contacts with an operator F describing the work as noema of consciousness (Husserl's noema), after the contact a certain reduction of state is occured, which may correspond to the noesis (Husserl's) of consciousness. A part of the reduced state is stored in brain as a memory. The scheme of our model is represented in the following figure.

87

O

Processing regions channel <=» evolution, gate in q.computer +a(dissipative, amplifying in C memory stored, pre-consciousness

Let us further assume the Hilbert space "Ho is composed of two parts, before and after recognition. For notational simplicity, we denote the Hilbert spaces by Hi,Hi,H3 where HI represents the processing part, H% the memory before recognition and Hs = H? the memory after recognition. Throughout this paper we will have in mind this interpretation of the Hilbert spaces Hj (j — 1,2,3). However, this is just an illustration of what we are going to do, and the teleportation scheme may be applied to very different situations. We are mainly interested in the changes of the memory after the process of recognition. For that reason we consider channels from the set of states on H\®H-2 into H%. Main object to be measured causing the recognition is here assumed to be a self-adjoint operator

F= k,l=l

on Hi®H.2 where the operators Fk,i are orthogonal projections (alterna-

88

tively, we may take Fk,i as an operator valued measure). The channel Afc,; describes the state of the memory after the process of recognition if the outcome of the measurement according to F was zk,i and is given by Ak

where p and 7 (denoted po above) are the state of the processing part and of the memory before recognition and J an isometry extending from HZ to H'2®'H.3 and I denotes the identical operator. The value Tri )2 ,3(^fc,;®I)(/>®'/7J*)(-Pfc,(®I) represents the probability to measure the value zk,i. So, obviously, we have to assume that this probability is greater than 0. The state Afc]j(/o(8»7) gives the state of the memory after the process of recognition. The elements of a basis (&fc)JJ =1 of "Hj are interpreted as elementary signals. In this first attempt to our model described above, there appear still a lot of effects being non-realistic for the process of recognition. Some examples (cf. the last section) show that with this model one can describe extreme cases such as storing the full information or total loss of memory, but - as mentioned above - that is still far from being a realistic description. In this paper we restrict ourselves to finite dimensional Hilbert spaces. Moreover, we assume equal dimension of the Hilbert spaces "Hj (j = 1, 2, 3). It seems that infinite dimensional schemes will lead to more realistic models. However, this is just a first attempt to describe the brain function. Moreover, for finite dimensional Hilbert spaces the mathematical model becomes more transparent and one can obtain easily a general idea of the model. To indicate obvious generalizations to more general situations and especially to infinite dimensional Hilbert spaces we sometimes use notions and notations from the general functional analysis. In a forthcoming paper we will discuss a modification of the model using general splitting procedures on a Fock space4'5. We hope to be able to include more realistic effects. Acknowledgement: The first two authors very much appreciate the support of the research by INTAS Project 99/00545. 2. Basic Notions Let 7^1,7^2,7^3 be Hilbert spaces with equal finite dimension: d i m W j = n , ft e {1,2, 3}).

89

First we will represent these Hilbert spaces in a way that it seems to be convenient for our considerations. Each of the spaces T~ii,H2,%3 can be identified with the space Cn of n-dimensional complex vectors. The space C" again may be identified with the space {/ : G —> C} of all complexvalued function on G := {1, . . . , n}. The scalar product then is given by =

k=i

J

f(k)g(k)n(dk)

where // is the counting measure on G, i.e. \JL = X)fc=i ^fc with 6k denoting the DlRAC measure in k. So, each of the spaces T~tj can be written formally as an L2-space: •Hj=L2(G,ri:=L2(G)

(j € {1,2,3}).

For the tensor product one obtains f®g(k, 0 = f(k)g(l)

(/, g 6 L 2 (G), k, € G),

and we have Hi^tHz = L2(G x G, /i x n) = W 2 ®ft 3 . We will abbreviate this tensor product by L^G 2 ,/^ 2 ) or just by Z/ 2 (G 2 ). By BCH) we denote the space of all bounded linear operators on a Hilbert space "H. In #(L 2 (G)) the operator of multiplication by a function g £ L%(G) is given by (Ogf)(k)=g(k)f(k)

(/

Observe that for all /, g €. L 2 (G) one has

and for / e L 2 (G) with f ( k ) ± 0 for all k £ G it holds Ojl = O1/f. The function 1, l(fc) = 1 for all k 6 G, obviously belongs to L 2 (G) and I = £>i is the identity in B(L2(G)). Consequently, an operator of multiplication Of is unitary if and only if |/(fc)| = 1 for all k 6 G. Further, we will use the mapping J from L2(G) into L 2 (G 2 ) given by (J f ) ( k , l ) = f(k)6k,t

(/6L2(G),fc,JeC)

(1)

where 5^^ denotes the KRONECKER symbol. It is immediate to see that J is an isometry. For the adjoint J* : L 2 (G 2 ) —> L 2 (G) we obtain = $(fc, fc) ($ e L2(G2), fc e G).

(2)

90

Observe that G equiped with the operation ® : G x G —> G, k © I := (k + l)mod n is a group. The operation inverse to ® we denote by 0. Let us remark that k © I = k — I'm the case k > I and kQl = k — l + n if k < I. We conclude that for all k € G the operator Uk € B(L2(G)) given by (Ukf)(m):=!(k®m)

(/ £ L 2 (G))

(3)

is unitary. Now, let (6fc)£=1 be an orthonormal basis in L/2(G), and denote by (Bk)%=1 the sequence of multiplication operators corresponding to the elements of this basis, i.e. Bk := Obk , k 6 G. Lemma 2.1. For k,l £ G we put

&,r.= (Bk®Ui)J 1.

(4) 2

The sequence (£k,i)k,ieG is an orthonormal basis in L^G ). Proof: First observe that for all k, I £ G we have £k,i(m,r} = bk(m)6m
(m,r £ G).

(5)

So one gets

<&j,&,J> = / J

^,i(m,r)-^k,i(m,r)^(d[m,r})

Since j ^ / implies ^m,rej^m,r©; = 0, the right side will be equal to 0 in this case. Further, observe that for all l,m € G there exists exactly one r e G such that r ®l = m, namely r = m — I if I < m and r = m + n — / i n the case I > m. So we may continue the above chain and get for the case 3=1 (&,i,£k,i)= I I

bi(m)bk(m)6m,r®ip2(d[m,r})

= I bi(m)bk(m)p(dm) = (bi,bk) Consequently, (^fc,;)fc,ieG is an orthonormal system in L^G2), and since dim L 2 (G 2 ) = n2 it is a basis in L2(G2). D We denote by Fitj G B(L,2(G2)) the projection onto &J, i.e. ^•:=l^){^l = <^v)^.

(6)

91

Remark: Sometimes (especially in proofs) the 'scalar product' notation is more convenient, but in some other cases using the 'bra-ket' symbols the statements become more transparent. So we will use both descriptions in the sequel. Observe that for $ e 1/2 (G2) and i, j € G one obtains

Indeed, we get from (5) and the definition (6)

= £i,j I I

= 6,j

bi(u)5u^&j^(u,v)^(d\u,

bi(

In Section 5 we investigate concrete teleportation channels. For this we need explicit expression for the operator (fi,j®I)(I®-/)- Using the definition (1) of the imbedding operator J and (7) we obtain for all k, I, m £ G

v=l

what leads to

®j,m)

(8)

2

for all $ 6 L 2 (G ) and i,j,k,l,m£ G. Now, we put for i,j g G Giti := J*(^-®I)(5r®I) = J*(C/,-5*®I)

(9)

where B* = OJ. = £>E.. For $ e L2 (G2) and m e G we get

The linear operator GJJ maps from Li(G2) into 1/2 (G) (it is not an isometry), and we finally get for $ £ Z»2(G2) (Fij®I)(I(gi J)$ - ^®Gij*. Example 2.1. Consider the orthonormal basis (6fc)fc=i i-2(G) given by Afc(m) = 6k,m. From (5) we get

(10) =

(^fe)fc=i

m

92

i.e. £itj = A»®Aie and we obtain for $ € (Gi,j$}(m) = bi(m 0 j)$(m 0 j, m) = A iej (m)$(i, i 0 j). Summarizing, in this special case we have

j-

(11)

3. Entangled States Definition 3.1. Let 7 be a state on W2 = Lz(G) (i.e. 7 is a positive traceclass operator with TV (7) = 1). The state 6(7) on Z/ 2 (G 2 ) = Hy^'Ha given by e(7) = J^r

(12)

where J is the isometry given by (1) we call the entangled state corresponding to 7. Example 3.1. Consider the basis (&fc)£ =1 = (Afc)5J=1 defined in Example 2.1, and let 7 be the pure state \(l/^/n)\ >< (l/v/n)l| (we recall that (l/Vn)l(fc) = 1/Vn for all k e G). For each observable A e B(L2(G) one has Tr(7yl) = nSfe=i-^^(^)- Especially, the quantum expectation of a multiplication operator Of, f e L^(G} will be just the arithmetic mean:

Observe that (1/v/n)! = (1/V") Em=i A m and for all $ e L 2 (G 2 ) it holds (Am, J*$) = (Am
l

, ~ m< ^ m , $ ) -7= E Vnm=l / Vnm=l

A

«i® A m(k, 0

Consequently, 0(7) is the state on £ 2 (G 2 ) = W2®Hs given by 1 vv

" m=l

1 >< - = v

n

AmAm m=l

This state 6(7) is a special representation of the entangled state used for the elementary teleportation model 2.

93

Now, let p and 7 be states on ~H\ resp. 7^2, the state 6(7) (usually denoted by a 6 ) will be a state on 'Hz&Hs. . Remember that we assumed HI = T~i2 = H.3 = I/2(G). The numbering only indicates the meaning of the states (we recall that HI represents the processing part, 7^2 the memory before and "Hs the memory after the recognition process.) Then p®e(-y) is a state on 'Hi®'H2®'H,3 and we observe immediately />®e(7) = (I® J)(p®7)(I® J*).

(13)

In Section 5 we calculate explicitly the trace of (FiJ®I)(/3®e(7))(Fi,j®]I) = (FiJ-®I)(I®J)(p®7)(I®J*)(Fij-®I). (14) The following proposition will be very useful for this. Proposition 3.1. Let (gk)^=i and (hk)^_1 be orthonormal systems in L2(G) and p and 7 states on L-z(G) having the following representations: hk\, fc=l

k=l

ak > , / f c >

fc=i

fc=i

Then for all i, j 6 G

fe,i=i where Gi^ is given by (9).

Proof: Using especially (10) we obtain for i,j,k,l 6 G and /i,/2,/s

Consequently,

.,• ® I) (I®J)

E a /3 (g ®hi,-)g ®hi k

fe,i=i

t

k

k

94 n

= Y,

akpi(Fi,j®l.)(&®J)(gk®hl,')gk®hl(1L®J*)(Fi,j®V)

k,l=l n

= Y,

a

kPiFij®(Gi>jgk®hi,-)Gitjgk®hi

fc,Z=l

= Fitj fc,J=l

what ends the proof.

D

Example 3.2. Let us return to Example 2.1, and suppose p and 7 are given as above but with gk = hk = Afc. Then

k,l=l

k,l=l

Consequently,

4. Channels Denote by T the set of all positive trace-class operators on L^G) including the null operator 0,

0(/) = 0

(/ e L 2 (G)).

We fix an operator r £ T having the representation n

< hk\

(16)

with (jk)k£G C [0, oo) and (/ifc)fceG being an orthonormal basis in Lz(G). The linear mapping KT : T —> T given by n

KT(p):=^>ykOhkpO*hk

(pzT)

(17)

depends only on the operator r but not on its special representation. Indeed, the following lemma holds Lemma 4.1. Let T have besides (16) a second representaion >
k=i

95

with (Pk)kea C [0,oo) and (<7fc)fceG being an orthonormal basis in For arbitrary p € T , it holds P0*gk.

fc=i

(18)

fe=i

Proof: It suffices to show (18) for p e T of the form p = \f >< f\ with / <E £ 2 (G). Since

fc=i

one obtains

fe=l

n

Jk(hk,

-}hkO} =

, -)gkf = 2(3kogkPo*gk. fe=i

n Definition 4.1. Denote by 5 the set of all states on L2(G) and for r e T by «ST the set of all states /9 from S with the property that TrKT(p) is positive: «ST := {p 6 5 : TV^r(/9) > 0}.

(19)

For r e T the mapping E"r : <Sr —> <S given by

^r(p)

(pe5 T )

(20)

is called the CHANNEL corresponding to r. The channel corresponding to T is called UNITARY if there exists an unitary operator U on Lz(G) such that KT(p) = UpU*

Observe that the channel KT is in general nonlinear. However, in Examples 4.3 and 4.4 below the channels are even unitary. Let us make some remarks on the physical meaning of the channels Kr and KT . The channels KT are mixtures of linear channels of the type Kh(p) := OhP0*h

(p € T)

96

with h 6 L2(G), \\h\\ = 1. Let us consider the more general case \\h\\ >0,

\h(k)\
(fceG).

We define an operator th : £ 2 (G) —> L2({1,2} x G) by setting for all / € L 2 (G) andfce G (h(k}f(k) for i = 2.

The operator th is an isometry from L 2 (G) to L 2 ({l,2}xG) = L 2 ({1,2})<8> L 2 (G). Indeed,

ii*/. /ii 2 = E E i*" /(*•fc)i2= E (iM*)i2 + 1- iMfc)i2) i/wi 2 = ii/ii 2 . 1=1 k=l

k=l

Consequently, the mapping Eh : B(L2({1,2} x G)) —> B(L^(G)) given by Eh(B) := t*hBth is completely positive and identity preserving. The channel E^(p) is the corresponding linear channel from the set of states on L 2 (G) into the set of states on L2({1, 2} x G). The space L2({1, 2} x G) has an orthogonal decomposition into L2({1} x G) and L2({2} x G) both being trivially isomorphic to L 2 (G). Performing a measurement according to the projection onto L2({1} x G) = L 2 (G) given the state E^(p) one obtains the state Kh(p). A measurement according to the projection onto £2({2} x G) = L 2 (G) leads to the state ^v/1-W2(/0). Finally, let us mention that from the statistical point of view one could get a deeper insight by considering the second quantization of that procedures. This means especially to replace pure states by the corresponding coherent states and the channel E£ by the corresponding beam splitting 5. Example 4.1. Assume h € L2(G) fulfills mfk€o \h(k)\ > c> 0. Then the function g € L 2 (G) given by

fulfills the above conditions \\g\\ > 0, \g(k)\ < 1 for all k e G and we obtain for all states p on L 2 (G)

97

Example 4.2. The identitiy r — I, (i.e. r(g) = g for all g € Li(G)) can be written in the form T = J^tec l#fc >< 3^1 where (fffc)fceG is an arbitrary orthonormal basis in L<2(G). Now, let p be an arbitrary element from T, p = ^k£Gak\hk >< hk\ where (hk)ke.G is an orthonomal basis in J and (afc)fceG C [0, oo). Then KT(P) =

Because of

the chain may be continued and we obtain

where iec

If /9 belongs to ST then 1 = X),-eG a}• = X),-cc 7j and

Consequently, (UJ}J^G is a probability distribution on G and the channel Kr(p) = KT(p)

transforms each state p € Sr into the corresponding "classical" state. Example 4.3. If T is the pure state corresponding to (l/i/n)l (cf. Example 3.1) one gets KT(p) = ( l / n ) • p for all p e T. For p & Sr one obtains Kr(p) = P.

98

Example 4.4. Suppose T = \b X b\ where b e L^(G} satisfies \b(k)\ = n for all k 6 G (in Example 4.3 we assumed b(k) = I/A/™)- Since ) = (l/n)Tr(p) we obtain for all p 6 5 KT(p) = BpB* with B = \fnOb. As we remarked on page 5 Oh is unitary if and only if \h(k)\ = I for all k. Consequently, the channel is unitary. If T is a mixed state KT(p) usually will be mixed even for pure states p. Below we give a simple example for this. Example 4.5. Let (A m ) m6 G denote as in Example 2.1 the basis in L2(G) given by A m (/c) = 6m>k. Put r = ^{Ai, ->Ai + s{A2, -)A 2 and let p be the pure state corresponding to -4g(Ai+A 2 ). Then KT(p) = \r and KT(p) = r. Example 4.6. Let T = {/,•)/ be a pure state, and assume / fulfils f(k) ^ 0 for all k. Consequently, I// e L 2 (G) and for all p e S = ST Ol/fOfpO*fOl/f

= p.

This implies K~l — Kf with f = ( j , - ) j - Normalizing f to a state one could write alternatively K~l = \2K?, T = (g,-}g with g = - • 4 and

5. The State of the Memory after Recognition

Let us recall that for states p, 7 on 1/2 (G) and i,j G G

is a linear operator from L2(G3) into L 2 (G 2 ), and that (cf. (14)) it is equal

In the following we consider the family of channels (Ajj)ij 6 G from the set of product states /9<8>7 on Wi®H 2 into the states on 7^3 given by

where Tri^ resp. Tri^.a denotes the partial trace with respect to the first two components resp. the full trace with respect to all three spaces. In the sequel we always will assume that > 0.

(22)

99

Let p and 7 are given as in Proposition 3.1. Since (£i,j)i,jeG is an orthonormal basis in Z/2(G 2 ) (Lemma 2.1) we get from Proposition 3.1 ) G^gk®hi (23) k,l=l

Summarizing, we get the following representation of AJJ: Proposition 5.1. Let p and 7 be given as in Proposition 3.1. Further, assume (22). Then (24)

Z^k,l=l l*kPl\\^i,jyk(&i<'l\r

where for $ € C^G^) n

'

'

'

(25)

m=l

Example 5.1. Let /9 and 7 be pure states, p = (5, -)ff,7 = (/i, -)/i. Then

Fortunately, we can find expressions for the state Ajj(/9(8>7) of the memory after the recognition process being in many cases simpler. We can express the teleportation channel AJJ with the help of the channels Kr we introduced in the previous section. Proposition 5.2. Let i, j e G and let p be a state from S,^r><^:, (cf. (17) and Definition 4-1)- Further, let 7 be a state from S such that eSr.

(26)

Then Aij (P®j) - #7 o K* o KK><^ (p)

(27)

where Ri denotes the unitary channel given by K^(p) = UjpUj . Proof: Let p and 7 be given as in Proposition 3.1. We set Bi = O^. Thus B? = %. From tf i^xST, fc) = B^pBi we conclude & o K^^p) = UBpBiU* what leads to

100

Ky 0 & o

><57(p)

= /=! n

k,l=l n k,l=l n

^

akPi(r(UjB;<S>V)gk®hi,.)J*(UjB';®V)gk®hi

k,l=l

i,-} Gitjgkhi k,l=l

Finally, from jL/i^'-'^.L'i | "'

we obtain (27).

>') = A~' o A J o A,—

D

In the following let us comment the results and give some examples. Let p be an arbitrary state of the processing part (the brain), and assume the measurement of the incoming signal leads to the value z i t j. Then the input in the memory being in the state 7 will be C

D* PJLJ^U-• A* D TT* • TT \J A±j-

where C is the normalizing constant. After the recognition process the brain will be in the state W o^ \(Jjt>4 (TT D*«D 7T*\] C • x\-y PD^UA

/OQ\ \^o)

where C is again the normalizing constant. Example 5.2. Let us consider the extreme cases that either the processing part or the memory is in the trivial state ,.. ._ X .—

-~,,^ -I I _ ±1 >< —=L\

V"

/OQ-l

\£")

\fn

(cf. Example 3.1). This state has no experience, no special knowledge, there will be no selection of incoming information. It is easy to check that for all n 6 S it holds Kx(n] = A*

(30)

101

(observe that KK(n) = ( l / n ) • /j,). On the other hand, for all states fj, the relation

is true. Now, we consider the case of the memory being in the state 7 = x. We obtain from (30) and from (28) for all i, j € G and p e «Sij:><(n

The memory will store exactly what comes in (the system is able to learn everything - cf. also 6>7 ). Since Uj and \JnBi are unitary operators (G = n) we see that for all i,j there exists (in the language of teleportation procedures) a unitary key Vij to recover p, i. e. A^ j (/0®7) = VijpV*jNow, let the processing part (the brain) be in the state p = x defined by (29). Then

For all states of the brain 7 such that \Ujbi >< Ujbi\ 6 iS7 we obtain

So (as one could expect) the final state in the memory depends only on the measured value Zjj and the state of the memory (before recognition). Example 5.3. Let (&j)j€G be an orthonormal basis fulfilling

For the pure state p = i I we obtain for all i 6 /

/T\

\

IbiXbil

n

l b il

1

n2

\—"^

A

Z^

Consequently, for j & G

For each state 7 (we use notation as in Proposition 3.1) and all i, j e G we finally get

102

with

=E

Pk\hk\2(l) n

So we obtain a classical state with probability distribution Example 5.4. Take (bj)j&o as in Example 5.3 above, but now suppose 7 = jjl. Let /? be given as in Proposition 3.1. For all i, j 6 G we get

*'' ° %><5T| G») = UjBtpBiU] = £ cek\UjB:gk)(UjB;gk\.

(31)

fc€G

Observe that for all i, j the sequence (UjB*gk)k&G is an orthogonal system. Indeed,

Consequently, as in Example 4.2 and above we conclude

p) = E 7r°Mfl?»i' E fc€G

ak n ^2 U I^S*I 2

-^E feSG

with

^= Finally we thus get

Ai,j(/o ® 7) = fc6G

with fcec We see that A^j does not depend on i. Especially, there do not exist unitary keys.

103

Example 5.5. We take again (bj)j&c as in Examples 5.3 and 5.4. Let us further assume that 7 = \h >< h\ is a pure state satisfying \h(j)\2 = ^ for all j € G. Using (31) we get for arbitrary p K7 o Ki o tf|5:><57| (p) =

OhUjBtpBtUjOt

Observe that \fnOh and \fnBi are unitaries. Consequently, we get A<j(/o®7) = ^-pV;*,with the unitary key V^j = nOhUjB*. The choice of the basis (6fc)fcec is very important in this model. Because of the specially chosen projection operators Fitj these are the only elementary signals that can be measured. Let us consider the case that the selected basis (&fc)fc=i is given by (Afc)£ =1 . In this case we get an especially simple (but also trivial) output. Let us remark that for all r, k, I 6 G such that k ^ I it holds Afc(r)A;(r) = 0. Thus the elements of the basis fulfil a condition much more stringent than just being orthogonal. Example 5.6. In Example 2.1 we obtain for p = X^=i ak\^k >< Ajt|, 7 =

and if oij > 0, PiQj > 0

So if OLi > 0, fliQj > 0 the state after recognition depends only on the measured value i Q j. If p resp. 7 cannot occur in the state Aj resp. AJQ., no information about the input can be stored in the memory. Example 5.7. Let p = Y^k=iak\9k >< 9k\, 7 = ELi^l^ >< "M be arbitrary states. What will be the state of the memory after recognition if only these elementary signals A^ can be measured? Will these elementary signals also damage the states p and 7 in such a way that only the information whether on > Oand f3iQj > 0 plays a role? Simple calculations as above lead to the following form (in what follows we omit normalizing constants): n

^ akftlAkj >< Ak,i\ k,l=l

104

where Ak,i(r) = &iQj(r)gk(r ® j)ht(r)

(r 6 G).

Now, for each k e G there exist sequences (/4)™=i> (^fc)?=i sucn that Sfc = £?=i MfcA, andfcfc= £?=1 "IA*' This implies 4*,, = ^ejAiej. Consequently,

k,l=l

The state a = A» j (p<8>7) after recognition will be the same as in the above example. We see that measuring Zij the state a will be able to store in his memory at most the signal A.iQj . And this can be done only if there exists at least one pair (k, 1) such that afc/?;!/^^6^2 > 0..

Concluding remarks: The aim of the paper was to touch the problem of finding simplified models for the recognition process. We were interested in how the input signal arriving at the brain is entangled (connected) to the memory already stored and the consciousness that existed in the brain, and how a part of the signal will be finally stored as a memory. Just to achieve simple explicit expressions we illustrated the model on the most simple sequence of signals (Afc)jJ =1 . It is clear that this example is just for illustration and can not serve for describing realistic aspects of recognition. Choosing a more complex basis one obtains expressions depending heavily on the states p and 7. Though the above presented model is only a first attempt it shows that there are possibilities to model the process of recognition. To get closer to realistic models we will try to refine the above models by -

passing over to infinite Hilbert spaces, replacing pure states by coherent states on the Fock space, considering different Hilbert spaces 'Hi^'Hz and T-iz, making more complex measurements than simple one-dimensional projections Fij, - replacing the trivial entanglement J by a more complex one based on beam splitting procedures, and finally - adding an entanglement between the states p and 7 on "H\ and H<2.

105 References 1. L. Accardi and M. Ohya, Compound channels, transition expectations, and liftings, Appl. Math. Optim. 39, 33-59 (1999) 2. L. Accardi and M. Ohya, Teleportation of general quantum states, quantph/9912087 (1999). 3. C.H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres and W.K. Wootters, Teleporting an unknown quantum state via Dual Classical and EinsteinPodolsky-Rosen channels, Phys. Rev. Lett. 70, 1895-1899 (1993) 4. K-H. Fichtner and W. Preudenberg, Characterization of states of infinite Boson systems L- On the construction of states, Commun. Math. Phys. 137, 315-357 (1991) 5. K-H. Fichtner, W. Freudenberg and V. Liebscher, Time evolution and invariance of Boson systems given by beam splittings, Infinite Dimensional Analysis, Quantum Probability and Related Topics 1 no. 4, 511-533 (1998) 6. K-H. Fichtner and M. Ohya, Quantum teleportation with entangled states given by beam splittings , Commun. Math. Phys. 222, 229-247 (2001) 7. K-H. Fichtner and M. Ohya, Quantum Teleportation and Beam Splitting, Commun. Math. Phys. 225, 67-89 (2002) 8. M. Ohya, Mathematical Foundation of Quantum Computer, Maruzen Publ. Com., 1999. 9. M. Ohya, Complexity in quantum system and its application to brain function, Quantum Information II (ed. T.Hida and K.Saito, published by World Scientific), 149-160 (2000) 10. M. Ohya, I.V. Volovich,Quantum Computers, Teleportations and Cryptography, Springer-Verlag, to be published.

PREDICTION ERRORS AND COMPLETELY POSITIVE MAPS

ROLF GOHM Universitat Greifswald, Institut fur Mathematik und Informatik, Priedrich-Ludwig-Jahn-Str.l5a, D-17489 Greifswald, Germany, E-mail: gohm@mail. uni-greifswald. de Abstract. We introduce the concept of an adapted isometry which is an operatortheoretic characterization of the time evolution of a stationary stochastic process adapted to a filtration. Using a product decomposition of an adapted isometry it is shown that prediction errors with respect to the filtration correspond to a sequence of completely positive maps. Asymptotic properties of this correspondence are studied. In a special case the computations can be simplified by stochastic matrices.

1. Introduction In this paper we want to describe a surprising link between prediction errors for a stationary stochastic process and completely positive maps on corresponding spaces of operators. There is no Markov property involved and the theory applies to stationary processes in general. These are much less treated than Markov processes and there is a lack of good techniques to do that. The above mentioned result indicates that combining the stochastics with certain operator-theoretic concepts provides a technique which makes at least some features of stationary processes more accessible. Let us describe the setting in more detail. We consider discrete time steps of a stationary process which is adapted to a filtration. Prediction in classical probability means to compute the conditional expectation of a random variable of the process with respect to a sub-cr-algebra in the nitration corresponding to some earlier time. From stationarity there is an isometric time shift on the associated L2-space. In Section 2 we introduce the concept of an adapted isometry which captures the essential features of such a time shift with respect to the filtration in a purely operator-theoretic way. Then a product decomposition of an adapted isometry is presented which makes possible the step-by-step analysis to follow. In Section 3 we use these structures to study the prediction problem. 107

108

We define a sequence of completely positive maps acting on spaces of trace class operators in such a way that certain non-linear prediction errors of the process with respect to the filtration can be expressed by products of these maps. In other words, the evolution of the prediction errors in time can be described by a sort of dynamics which is well known in quantum theory as an irreversible dynamics of mixed states. This correspondence is our main result in this paper. As a first application we relate determinism of the stochastic process to asymptotic properties of this quantum dynamics, namely absorbing vector states. Loosely stated: The mathematics behind the property of determinism for a stationary process can be translated into the mathematics describing the return of an excited atom to its ground state. To focus on the main idea without technicalities we give in Section 4 a detailed discussion of a simple case: commutative stationary processes with finitely many values. Now the spaces of trace class operators are finite dimensional and with respect to a suitable basis the completely positive maps turn out to be stochastic matrices. This allows us to write down purely combinatorial versions of our results. Using well known facts about stochastic matrices the asymptotic behaviour is then quickly determined. It should be emphasized however that the familiar reasoning with stochastic matrices in Section 4 depends ultimately on the operatortheoretic results given before. So we have a situation where putting the stochastics into an operator-theoretic setting provides new insights. In fact, our theory applies in the same way to commutative and to non-commutative stochastic processes as studied in quantum probability. We indicate these possibilities at suitable places in the text but to maintain the introductory character of this paper we decided to postpone detailed applications to non-commutative processes and to operator algebras to later work.

2. Adapted isometrics Let us first mention 7 as a general reference for operator theory concepts occurring in this paper. We consider Hilbert spaces which are all assumed to be complex and separable. Let {/Cra}J£L0 be a sequence of such Hilbert spaces and let {On}^L0 De a sequence of unit vectors so that fJ n 6 K-n for all n. Then there is an infinite tensor product K. — <3CL0 ^™ along the given sequence of unit vectors (cf 7, 11.5.29). There is a distinguished unit vector fi = <Sdo &n £ &• Further we consider the subspaces /C[min] = <S>"=m £j (m < n) of 1C where 77 6 K[m
109

(g)™^1 flj ® r? ® ®£ln+i fij 6 £• Then /C is the closure of U^°=0 ^[o.n] • An operator a £ #(/C[m,n]) (B denotes bounded linear operators) is identified with I[o,m_i] ®a® I[n+i)00) € B(JC). Definition 2.1. An isometry v € B(JC] is called adapted (with respect to ] C /C[o >n+ i] /or a/Z n e N0 and vQ = fi. This terminology is motivated by probability theory. Let us indicate how adapted isometries arise from stochastic processes: Let (fi, E, //) be a probability space and {Sn}^L0 a sequence of independent sub-tr-algebras. Denote by S[m,n] the cr-algebra generated by all EJ for m < j < n. In particular we get a filtration {E[o>n]}^Lo °f increasing sub-u-algebras. We assume that they together generate S. There is a (discrete time) stationary stochastic process on (fi,S,/z) determined by the sub-cr-algebra S0 (representing time 0) together with a (not necessarily invertible) measurable and ^-preserving transformation T : fi —» fl. Namely, functions which are measurable with respect to f~™So may be interpreted as random variables of the process at time n. The process given by EQ and T is adapted with respect to the filtration {E[0i7l]}£l0 if f~"S 0 C £[o,n] for all n. This is the case if f~ 1 E[ 0)n ] C S[0,n+i] for all n, and it is no serious restriction of generality to consider only this setting. Because f is /u-preserving it induces an isometry v on the square integrable random variables by v£(w) := £(TLJ) for £ e L 2 (fi, E,/x). Assume that L 2 (fi, E,/i) is separable as a Hilbert space and that f^Ejo^] C E[o >n +i] for all n. Then it is easy to check that v is an adapted isometry with respect to {L2(0, S n ,^), ln}%L0, where l n denotes the constant unit function considered as an element of L 2 (0, S n , p.): First use the well-known fact that independence of cr-algebras implies a tensor product decomposition of the corresponding L2-spaces and then translate the properties of f into properties of v. Let us remark here that any adapted isometry may be constructed by probabilistic means if we include non-commutative processes in the sense of l . Here the probability space and its sub-cr-algebras are replaced by a unital *-algebra with a state and its unital *-subalgebras, the transformation f by a unital *-homomorphism preserving the state and the space L 2 (fi, S, /x) by the corresponding GNS-Hilbert space. If we then use a notion of independence based on tensor products then we may repeat the arguments above in this more general setting. In particular if we start with an adapted isometry v as in Definition 2.1 we may use B(JC) as an algebra, J7 as a vector state, subalgebras B(K.^m^) and the homomorphism defined by x \—> v x v* .

110

From this we can reconstruct v as the adapted isometry corresponding to a non-commutative stationary stochastic process. Because we shall not explicitely consider non-commutative stochastic processes in this paper we do not give more details. Let us now analyse the structure of an adapted isometry from an operator-theoretic point of view. Proposition 2.2. Let a sequence {^Cn>^n}£ 1 un 6 S(/C[0in]), un£l = fi, for all n — > 2 unn IA^ lie [ 0 ] r v _ 2 j = I |lie J A^ [ 0 , n _2]', then we can define an adapted isometry by v := stop — lim u\u-2 . . .un n—»oo (here "stop" denotes the strong operator topology). Conversely if dim tCn < oo for all n then any adapted isometry can be written in that way. Proof. If £ £ ^C[o,m-i] then it is fixed by any un with n > m + 1. Therefore v£ = lim uiu^ . . . un£ = uiU2...um£ £ ^CfomlBy approximation the l J '

limit exists for all £ € K-. To prove the converse let v be an adapted isometry. For all n > 1 consider the isometry vn : A3[o >n _i] —> ^[o,n] given by the restriction of v to /C[o,n-i]By assumption we deal here with finite dimensional spaces and therefore by dimension arguments there is an extension of vn to a unitary un £ B(/C[ 0>n ]). Now define un := u^_iUn (with UQ := I). We have un £ *B(/C[0,n]) and from vfl = f2 we also get unCl = £1. If n > 2 then because of un-i Ir"[0,™ — 2] = v \ic =un"\jc we find that u™n I"Mo,n-2] r — I Ir • D ' -[0,T,-2] 'V[0,n-2] '^[0,n-2] l/v

l V

l

We add some remarks. First: If we drop the condition of finite dimensionality for the converse direction then an inspection of the proof above shows that it may be necessary to enlarge the spaces ICn in order to proceed. This shows that any adapted isometry can at least be embedded into an adapted isometry of the product type above. Second: The condition un lie = I lir is implied by the more convenient condition '"-[0,^-2] '^-[0,71-2] un 6 B(K. [„_!,„]). It is an interesting question how representability with this stronger condition restricts the class of associated stochastic processes. We shall not pursue this question here. Instead for the rest of this paper we shall assume that the adapted isometry is given by a sequence of unitaries {un}^Li satisfying even the stronger condition and ask how we can use this r

J

111 step-by-step information to analyse the associated stochastic process. 3. Main results Let Ti.,K. be Hilbert spaces. Denote by T(. . .) the space of trace class operators with Tr the trace functional and Tr^ : T(H ® K,) -> T(/C) the partial trace obtained by evaluating the trace only on H . To any isometry v : H —> H ® 1C we can associate the operator Dv : T(H ) —> T(1C) given by p i—> Tr^ (vpv*). Then Dv is a completely positive and Tr-preserving map. In the physical literature such maps are important because they define time evolutions for mixed quantum states or density matrices (cf 25 ' ) . Mathematically this is the set of positive trace class operators with unit trace which we denote by Tj"(. . .). Because Dirac's notation is useful for the computations to follow we adopt it together with the physical convention of scalar products linear in the second component. Given a unit vector Q£ € K, we can consider Ji as a subspace of "H ® 1C by H ~ W <8> Q£ C H <8> K-- Let p be the corresponding orthogonal projection. We may interpret v as an isometric dilation of the contraction pv € B(H) (cf 6 , chapter VI). The following Lemma shows that Dv encodes information about the defect arising in this dilation procedure. Lemma 3.1. Assume £, £' € "H . Then

(b) Proof, (a) is immediate from the definition of Dv . To prove (b) we choose an orthonormal basis {e,} of 1C with ej = O^;. Then v£ = ^& <8> £i and > €i with {&}, {£} C W . We conclude that

*••» Lemma 3.2. Let W 0 , W i , W 2 6e ffi/6eri spaces, u : Ho®Hi unitary, w :Hi —^Hi^Hi isometric andv : defined as v = (u 1^ )(I-^ o iy). T/ien A, = Proof. For p € T(UQ <8> W i) we find that £>„, o

112

We want to use these results for the analysis of adapted isometrics. Let v e B(JC) be an adapted isometry (with respect to {>Cn, nn}£L0, see Definition 2.1) and let {wn}^Li be an associated sequence of unitaries as in Proposition 2.2. We assume that un £ B(£[ n _i >n ]) for all n. Now define isometries vn : /C n _i —> fc[n-i,n] = fcn-i ® K-n as restrictions of un to /C n -i- Using the procedure above we get a sequence of TV-preserving completely positive maps Dn := DVn : T(£ n _i) -> T(/C n ), n > 1. Note that Dn(\ fin_i > < fin_i |) =| fira >< fin (because u n fj = fi). For the subspace £[m,n] of K. let us denote by p[TOinj resp. Tr[m]Tl] the corresponding orthogonal projection resp. partial trace. Lemma 3.3. Assume £, £' e IC0. Then for all n 6 N0 Tr[0,n](\vn+lt'><^\) and !, Dn+1Dn . . . I>i(| £' >

Proof. The second part follows from the first by the same argument as in the proof of Lemma 3.1. To prove the first part we proceed by induction. The case n = 0 is given by Lemma 3.1 (note that v \£ = 1*1). Now for some n > 1 assume that Tr^.^d vn£' >< vn£ |) =°Dn...Di(\ £' > < £ |). We have vn+1£ = HI . . .unun+ivn£ = HI . . .unvn+ivn£ (and the same for £')• Inserting the Definition of Dv, then applying Lemma 3.2 With HQ = £[o,n-l]) ^1 = £n, T^2 = Kn+l, U = Ui...Un,W = Vn+l, V —

v |^

^ i and using the assumption we get

= Dn+1Dn...Di(\t'><£\).

D

Continuing our interpretation of Lemma 3.1 we may say that the product Dn+iDn . . . DI encodes information about the defect arising from the (n + l)-th power of an adapted isometry. This can be made more concrete by using probabilistic language. In Section 2 we gave an interpretation of an adapted isometry as a time evolution of a stationary stochastic process. Now we can interpret the term P[o,n]vn+1£ appearing in Lemma 3.3 as the best predictor of vn+l£, a random variable of time n + 1, given the information available up to time n. More precisely: the best non-linear one-step predictor in the mean square sense. "Non-linear" refers to the fact that not

113

only the linear span of random variables of the process is used for prediction but the whole algebra. A survey on several topics in prediction theory is 4. In our setting the (n + l)-th prediction error / n +i(£) (for f 6 IC0) is given by /n+1(0 := \\vn+1t -

n+1 t\\ P[0,n]v

= (\\t\\* - ||p[0,n]i5n+1f ||2)* •

From Lemma 3.3 we get the following formula for the prediction error: Theorem 3.4. For all £ € K,Q and n 6 N0 /n+i(0 2 + It is interesting that in linear prediction theory there is a similar formula for linear prediction errors which also involves products of certain quantities. In that case an additive decomposition of the Hilbert space is used to decompose the isometry. Its probabilistic interpretation is related to the iterative Burg's technique of prediction (cf 4, Th.5.1 or 6, II.5, II. 6). In our theory the Hilbert space is multiplicatively decomposed into tensor products and we have a corresponding decomposition of the isometry in Proposition 2.2. Trying to interpret the formula in Theorem 3.4 we observe that expressions like Dn+1Dn ...Di(\£ ><£ |) are well-known in quantum theory as an irreversible time evolution converting a pure state of a quantum system into a mixed one (cf 2'5). This opens up the possibility to use concepts from quantum theory for prediction problems. As a first application of this correspondence between prediction and quantum dynamical time evolutions we want to analyse asymptotics, i.e. the behaviour for large time (n —> oo). For this we need some technical facts about density matrices. For the convenience of the reader we also include an elementary proof for them. Lemma 3.5. Consider sequences {Kn} of Hilbert spaces, {fin} of unit vectors, {pn} of density matrices such that £ln S K,n, pn g T^(!Cn) for all n. Then for n —*• oo the following assertions are equivalent: (1) (2) || pn— |Q n >
114

we quickly infer (3) => ( 1 ) . It remains to prove that (1) => (2): Write pn = £c^n) e\n} Xe^ \ with a, 0, £>jn) = 1 i

i

and {e\ } an orthonormal basis of K.n. From ( J ) we get /

e,(

n

\

)^/^( n )| I n ^ _ ^.J™) !<-.>) o > , "i l^ti (''n^

If i = 1 is an index with o^ < e^ , fi n >

= max c*^ 2

for all n then because of

we infer aj"5 -> 1, i.e. £ c^n) -> 0 and

> i , i.e. Finally

\\pn- \an><Sln\

><en) -

<||ai n )

n Proposition 3.6. Assume £ e /Co, ||^|| = 1. TTie following assertions are equivalent:

(1) lim /n(0 = 0, >00

iim

Tl—>OO

(weak or with respect to the trace norm). Proof. Take the formula for / n (£) in Theorem 3.4 and then apply Lemma 3.5 with pn = Dn ... L Note that the sequence {/n(£)} of prediction errors is in any case a non-increasing sequence of non-negative numbers and thus there is always a limit /oo(£) '•— lim /n(0- This is immediate because the time interval used n—too for prediction increases and there is more and more information available. Proposition 3.6 gives a criterion for this limit to be zero, i.e. for prediction becoming perfect for n —+ oo. To formulate this criterion verbally we state some definitions.

115 Definition 3.7. (a) A stationary stochastic process given by /Co and an adapted isometry v is called deterministic with respect to the filtration {/Cn,nn}~=0 t//oo(0 := lim /„(£) = 0 for all £ € £„. n—>oo (b) If the conditions of Lemma 3.5 are fulfilled then the sequence {|fi n > 7"i"(/Cn). If {| On >< On )} is absorbing for all sequences {Dn(p}} with p € Tf (/Co) then we call {|fi n >< 0 | should be absorbing for the semigroup {Dn}^=0. Absorbing vector states for completely positive semigroups are a well-known subject in mathematics and physics (cf 3 > 5 ' 2 , terminology varies somewhat) and we make contact to it at this point. The problem can be approached via spectral theory for D and is closely related to ergodic theory. The inhomogeneous case is related by Corollary 3.8 to quantum evolutions with a time-dependent dynamics. This is more difficult and less treated. 4. Finite-valued commutative processes In this section we want to illustrate the theory developed above. The emphasis is on examples, we do not try to be exhaustive. As shown in Section 2 the concept of an adapted isometry includes commutative stationary stochastic processes. This raises the question how the results of Section 3 look like in this particular case. It turns out that

116

we can get an interesting view into the combinatorics of our approach by considering processes with finitely many values. Take {!,... ,d}®° (with some natural number d > 2) with the infinite product n of the probability measure giving equal weight to all elements of {1, . . . , d}. Then it is easy to check that a measure preserving transformation f of this probability space is adapted with respect to the natural filtration (i.e. gives rise to an adapted isometry when the construction in Section 2 is performed) if for all u = {un}%L0 £ {!,..., d}N° and all n € NO the values {(TO>)J}™=O depend only on {w;}"^1. Analogous to the argument given in the proof of Proposition 2.2 it follows that an adapted transformation T can be decomposed as an infinite product f = lim rn . . . r\ , n —>oo

where rn is a permutation of {1, . . . )C;}{°. •••.«} which acts identically on {!,..., d}{°'-'n~2} . Note that the value of (fuj)n is already determined by T ra+iTn • • -T\W, i.e. the limit is well defined. Our simplifying assumption that un 6 S(/C[n_i>Ti]) (see the second remark after Proposition 2.2) means here that rn is simply a permutation of {!,... ,d}^" 1>nK Now the prediction problem considered in Section 3 can be formulated as a game. If we are given only w0 , . . . ,wn of some w = {wn}^L0 6 {!,... ,d}N° then in general it is not possible to determine (f n+1 w)o. We may try to guess. It depends on f how much uncertainty we have to endure. Indeed the prediction errors show the amounts of errors (in the mean square sense) which are inevitable even with the best strategy. More precisely, let £ be any (complex- valued) function on {!,... ,d}. Given certain values LJQ , . . . , wn there is a probability distribution ^Wo ,...,<.,„ on {1, . . . , d} for (f n+1u>)0 conditioned by these values. Elementary probability theory shows that the best prediction of £((f" +1 w) 0 ) given w0, . . . , wn is obtained as expectation of £ with respect to /x a , 0j ... >Wn with the variance Var(£, /^ 0) ... iWn ) as squared error. Then the total mean square error /n+i(£) is obtained by averaging over all possible u0,...,wn: / n +i(£) 2 = Wo.....« This justifies the interpretation given above. In Theorem 3.4 we derived an alternative expression in terms of a product of completely positive maps Dm : Mj, —> Mj,. Here Mj, denotes the d x d-matrices (= T(Cd)). We have = \\t\\2.

Here Qn+i = ( l , l , . . . , l ) e C d . We want to write down the operators Dm more explicitely. For this consideration we drop the index. The operator D is derived from a permutation r of {1, . . . , d}2 giving rise to an isometry v : Cd -» Cd Cd so that for p e Md we have D(p) - TrCd(vpv*) (with

117

trace evaluated on the left). We want to calculate coordinates with respect to the normalized canonical basis {[i>}f = i of C d , i.e. i> has entry \fd at k position i and zero elsewhere. Let us write i —> j if the first component of r(i, fc) is j. Then a straightforward computation yields

Lemma 4.1.

v \j>=

-

£ \i> ® \k>

where jj counts the number of elements. Some observations about these coordinates of D are immediate: There is a symmetry Dki,ij = Diktji- Further, fixing k,l and summing over i,j always yields one, which proves the surprising fact that D with respect to the normalized canonical basis gives rise to a (row-) stochastic d2 x d2matrix. Its entries are a kind of transition probabilities for pairs when applying T, refining the transition probabilities for individuals which are k included as Dkk,u = U {r '• r —> i}. Putting all this together we have proved the following combinatorial formula which summarizes the computation of prediction errors in this setting: Proposition 4.2. For all n 6 NO, i € {1, . . . , d } :

d fn+l(\i>)2 + "I £ (Dn+lDn . . .Z?i)fc/,« = 1. k,l=l

The sum is a column sum of a (row-) stochastic d2 x d2 -matrix which is given as the product of the (row-) stochastic d2 x d2 -matrices associated to the operators Dm as in Lemma 4-1Of course the occurrence of stochastic matrices simplifies the asymptotic theory. See 8 for some basic facts about stochastic matrices, in particular: A set of indices is called essential for a stochastic matrix if by successive transitions allowed by the matrix it is possible to go from any element of the set to any other, but it is not possible to leave the set. An index not contained in an essential set is called inessential. Proposition 4.3. For the processes considered in this section the following assertions are equivalent: (1) The process is deterministic.

118

(2) All entries of the stochastic matrices associated to the products Dn ... DI which do not belong to an ii-column (i e {!,..., d}) tend to zero for n —> oo. And if the process is homogeneous (Dm ^ D for all m): (3) Indices ij with i ^ j are inessential for the stochastic matrix associated to D. Proof. Determinism means that fn(\ i', >) —-> 0 for all i € {1,... ,d} and n —> oo. By Proposition 4.2 this is the case if and only if the column sums ^2k!i=i(Dn+iDn • • --Di)fc;,ii tend to d. But the sum of all entries is d2 and none of the column sums can exceed d (obvious from Proposition 4.2). This proves (1) «=> (2). It is a general fact that for powers of a single stochastic matrix D we have the equivalence (2) «=> (3) (cf 8, chapter 4). D Especially condition (3) of Proposition 4.3 is very easy to check, at least for matrices of moderate size. We give an example: Choose d = 3 and consider the homogeneous process generated by the permutation T of {1, 2,3}2 given by the cycle (11,12,13,23,22,21,31,32,33). Using Lemma 4.1 we can compute the associated stochastic matrix. The result is shown below (with indices ordered as follows: 11,22,33,12,21,13,31,23,32) For example the non-zero entries in the fourth row (with index 12) are obtained from 1 -^-> 1, 1 -^-> 1 and 3 -^ 3, 3 -^ 3 and 2 -^ 3, 2 -^ 2.

'

1 3

_t

1 1 1 1 0 0 0

\J

1 2 0 0 0 0 1

\ o i

£i

\J

\J

1 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 1 0

\J

\J

\J

\J

0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 1 0 1 0 0 1 0 0 0 0 0 0 0

r\

f\

r\

It is easy to check that starting from any index ij we can in at most two steps reach the essential set {11,22,33}. With Proposition 4.3(3) it follows that the process is deterministic.

119

If we want to generalize to the inhomogeneous case then we may use the theory of inhomogeneous products of stochastic matrices (cf 8, chapter 3) to get explicit criteria for determinism. If the process has infinitely many values we can use infinite matrices or transition kernels in general. Note however that the stationary processes considered here are not Markovian in general and the probabilistic interpretation of these matrix entries is by no means obvious. They are derived from the completely positive maps defined in Section 3. If the stationary processes are non-commutative from the beginning then one has to work directly with these maps and the simplification obtained above for the commutative case is then not available. References 1. Accardi, L., Frigerio, A., Lewis, J. (1982). Quantum Stochastic Processes. Publ.RIMS Kyoto 18, 94-133. 2. Alicki, R., Lendi, K. (1987). Quantum Dynamical Semigroups and Applications, LNP 286, Springer-Verlag. 3. Arveson, W. (1997). Pure E^-semigroups and absorbing states. Commun. Math. Phys 187, No.l, 19-43. 4. Bucy, R. S. (1994). Lectures on Discrete Time Filtering, Springer-Verlag. 5. Davies, E. B. (1976). Quantum Theory of Open Systems, Academic Press. 6. Foias, C., Frazho, A.E. (1990). The Commutant Lifting Approach to Interpolation Problems, Birkhauser-Verlag. 7. Kadison, R., Ringrose, J. (1986). Fundamentals of the Theory of Operator Algebras, Vol. I, II, Academic Press. 8. Seneta, E. (1980). Non-negative Matrices and Markov Chains, SpringerVerlag.

THE Q-PRODUCT OF GENERALISED BROWNIAN MOTIONS

MADALIN GUTA Mathematisch Instituut, Katholieke Universiteit Nijmegen, Toernooiveld 1, 6526 ED Nijmegen, The Netherlands The notion of generalised Brownian motion is extended to multiple processes indexed by a set X. For — 1 < q < 1 the (/-product of positive definite functions on pair partitions having the multiplicative property is defined, and shown to be a positive definite function on X-coloured pair partitions. The resulting I-indexed generalised Brownian motion interpolates between graded tensor product (q = — 1), reduced free product (q = 0) and tensor product (q = 1) of the given Brownian motions.

1. Introduction Two different notions of independence stand out in non-commutative probability: tensor and free independence. The quantum stochastic calculus of Hudson and Parhasarathy 23 generalises the classical independence to tensor independence, and the free probability of Voiculescu 28 relies on free independence. Beyond the well established frameworks of these two theories, an investigation concerning the notions of quantum white noise, Brownian motion and Markov processes is developed by Kostler 17 in the spirit of Kummerer's approach to quantum probability 18>19>20. Related to this, is the theory of generalised Brownian motion initiated by Bozejko and Speicher which provides concrete examples of quantum white noises but also raises interesting operator algebraic questions. The generalised Brownian motions 4 are non-commutative processes w(f) indexed by elements / of a real Hilbert space K. and endowed with a distinguished "gaussian" positive linear functional pt given by the following pair prescription: ) 121

(fe.Oev

122

where the sum runs over all pair partitions of the ordered set {1, . . . , n} and t : UfeeN^ ) 2(2fc) —> C is a function defined on all possible pair partitions called positive definite. We regard u>(/) as symmetric operators on the Hilbert space arising from the GNS construction. The most important examples of positive definite functions are t(V) = 1 for all V which characterises the classical Brownian motion 24 and, t(V) = 0 for crossing partitions and t(V) = 1 for non-crossing partitions which characterises the free Brownian motion 28. A remarkable interpolation between these cases arises from the algebra of g-deformed commutation relations a(f)a*(g) — qa*(g)a(f) = ( f , g ) l for -1 < q < 1 investigated in a number of papers 5,e,7,8,9,io,i3,i4,2i,25,29 i The monomials in fields o>(/) := a(/) + a*(/) have vacuum expectations as in 1 with t g (V) = qcr(v^ and cr(V) being the number of crossings of the pair partition V. Further functorial and operator algebraic properties of this Brownian motion are studied in 3. It turns out that we have a functor of white noise 18'19 that is, a functor from the category of real Hilbert spaces with contractions to the category of non-commutative probability spaces. Another interpolation between the classical and free Brownian motion is found in 4 : t p (V) — pl y l~ B ( v ) where |V| is the number of pairs of V and B(V) is the number of connected components or blocks of V, and 0 < p < 1. Ispired by ideas from combinatorics, such as that of species of structures 1|1S and analytic functors 16, the author together with Hans Maassen n'12 have found a more functorial approch to the study of generalised Brownian motion. The GNS representation space of the algebra of fields w(/) over a real Hilber space 1C, with respect to the gaussian state pt has the Fock-like form 00

^®^c">

(2)

n=0

where Vn is a Hilbert space carrying a unitary representation Un of the symmetric group S(n), and s denotes the closed subspace of the tensor product Vn ® K.^n whose orthogonal projection is

and Unfl

<8> . . . ® /„ = /r-i(l) ® • • • ® /T-l(n)

(4)

123

for f i € K-c- The factor ^7 in 2 refers to the inner product on Vn ®s K,®n. We note that J^ is an endofunctor of the category of Hilbert spaces with contractions called analytic functor 16. The fields w(/) can be represented as the sum of creation and annihilation operators a(/) + a*(/) which are defined with the help of a sequence of operators jn : Vn —> V^+i satisfying the intertwining relations with respect to the representation of the symmetric groups Un and Un+i: jnUn(T) — Un+i(i(T))jn, with i the natural embedding of Un into Un+i by keeping the last letter fixed. Concretely, 0*(/) : Pn(vn ® Vn) "-» Pn+l(jnVn

® r(/)V»n)-

(5)

where vn S V^, Vn e £®n and r(/) is the right creation operator as defined on the full Fock space over K-c- For more details we refer to the first 3 sections of 12. Using this insight, a new class of positive functions on pair partitons has been found 2 , which extend the indecomposable characters of the infinite symmetric group and for which the gaussian state is not tracial. In the present work we answer a question of Roland Speicher concerning the existence of the ^-product of generlised Brownian motions. In his paper 26 Speicher has analysed the existence of universal products on the category of unital algebras with normalised linear functionals. The universal product should satisfy some natural requirements such as associativity and universal calculation rule for mixed moments. The result is that the only possibilities are the tensor product and the reduced free product. In the case of qdeformations, a concrete result in this direction has been proved in 21. However one can still define a ^-product for the algebras of generalised Brownian motions with gaussian states pta characterised by the positive definite functions t a with a in a index set X, by the calculational rule:

n £> The sum is taken over those pair partitions V such that if ( i , j ) € V then c(i) = c(j) € I, which we call the "colour" of the pair (i, j), the pair partition Va is the subset of pairs in V which are coloured in the colour a, and the coefficient cr(V, c) counts the number of crossings between pairs of different colours. The main objective of the paper is to prove the positivity of this functional for functions t a which have a certain multiplicativity property. This is done in section 3, but as a preparation for that we develop the framework for I-indexed generalised Brownian motion which is a fairly straightforward extension of the usual theory to the case of more than

124

one processes "coexisting", this time the key notion being that of positive definite function on Z-coloured pair partions. The (/-product interpolates between graded tensor product duced free product 2S (q=0) and tensor product 24 (q=l).

22

(q=-l), re-

Finally, a central limit theorem is proved showing that as n —> oo, the functional obtained by taking the ^-product of a function t a number n of times and restricting it to the algebra of fields o>(/) = -j= X)™=i w i(/)> converges in law to that of the g-deformed fields given by t g (V) = qcr^v\ 2. Generalised Brownian motions indexed by X and the "-semigroup BP^ (oo) Let 1 be an arbitrary index set. In this section we will extend the notions of generalised Brownian motion and the associated *-semigroup of broken pair partitions 12 to the case of multidimensional processes indexed by the set I. Definition 2.1. Let P be a finite ordered set. By P%(P) we denote the set of /-coloured pair partitions, that is pairs (V,c) with V e Py(P) an : £ « - > {!,..., \La\},

f^'.Ra^ {!,..., \Ra\}

(7)

are bijections. Any order preserving bijection a : X —» Y induces a map

«a : (Va, /«, /W) -> (a o V0, /<" o a-1, /« o a"1)

(8)

where a o V := {(a(i),a(j)) : ( i , j ) G V}. This defines an equivalence relation. The corresponding element d of BP^(oo) is such an equivalence class of collections of triples (V0, fa , f a ) for a e I. In other words the elements of BP^(oo) are broken pair partitions as in definition 3.1 in 12, with additional labeling with indices from J of the pairs and legs. For each a € 1 there is a broken pair partition d a , however only a finite number of them are not empty. The product and involution are defined as for the "-semigroup BP^oo) with the additional condition

125

that the left legs and right legs which are joined must be indexed by the same element of Z. Let di = (V a ,i, /$, /S)a6l and d 2 = (V0,2, /$, /^W be two elements of BP%(oo) with the notations from definition 2.2. Let M = min(|J?0ii , | £0,2!) be the number of legs which "join" by taking the product of di and d2. Then we define d i - d 2 = (V a ,/W,/W) o € l

(9)

with Va = Va,i U Va,2 U { (ttgr'W,

tfgr'W)

: i < M},

(10)

and the map /I defined on the disjoint union of the sets L 0) i of left acoloured legs from the left diagram di, and L0i2 \ (/ai2) * ({!,..., M}) consisting of the unpaired left a-coloured legs of the diagram d 2 by

The function /a is defined similarly. The product does not depend on the chosen representatives for dj in their equivalence class and is associative. The diagrams with no legs are the J-coloured pair partitions, thus P% (°°) C BPj(oo). The involution is given by mirror reflection. If d = (V 0 ,/i , fa ) then d* = (V*,/a , /a ) with the underlying set X* obtained by reversing the order on X and K := {(t,j) : CM) €E V0}

(11)

is the adjoint of Va. It can be checked that

(di • d 2 )' = d; • dj.

(12)

We are interested in positive functionals t on the "-semigroup which have the form otherwise.

(13)

The function t : 'Pj(oo) —> C will be called positive definite on T-coloured pair partitions and will be shown to characterise generalised Brownian motions indexed by 1. Let n : T —» N be a function which is equal to zero except a finite number of elements. We denote by BP%(n, 0) the set of diagrams d for which Ra = 0

126

and |L0 = n(a). The GNS-representation (xt,V,£t) with respect to t is characterised by

-d 2 )

(14)

which implies n

where

Vn = lin{Xt(d)& : d € BP$(u,0)}.

(15)

n

On Vn there is a unitary representation of the direct product group S(n) := X o6 iS(n(o))

(16)

each of the terms S(n(a)) permuting the a-coloured left legs of the diagrams in BPJ(n,0). We denote by ir(d) the diagram obtained by applying the permutation TT to the element d and the representation by Un. We distinguish the operators j0 := Xt(d 0 ,o) where da<0 is the diagram containing one left leg indexed by a £ I. The following intertwining relation will be important in later constructions: ja : Vn -> Va+Sa

ja-Un(r) = Un+Sa(ti\T)).ja

(17)

with (n + Sa)(b) := n(6) + 8a0 • (7r li2 , e)(da,0 • dii0 • d a , 0 )

(18)

where (TTI^, e) is an element of S(2) X S(l). The function t can be calculated as follows t((V, c)) = <&, fa fb j*a E/ 2 ,iK,2, e) ja jb ja & ) .

(19)

In general, let V £ PzCin) be a pair partition and c : V —•> I a colouring. Then c can be also seen as defined on {!,..., In} with the condition that

127

it takes the same value in points belonging to the same pair of V. We split the ordered set {!,..., 2n} in a number 2m of disjoint subsets B\ := { f c j _ i , . . . ,pi} and BJ := {pi + 1 , . . . ,fc^},with ko = 1 and kr = In such that the blocks B, contain left legs of pairs in V and Bj contain right legs. Finally the value of t can be written as the expectation t((V,c)) = /&, n ^ o t f n x f r l ) \

ft

(=1

Ml)

...tfnr(7Tr)

f=Pl + l

f[ f=pm + l

jc(l) & } .(20) /

On the basis of this structure we pass now to the construction of the generalised Brownian motion indexed by I. Let t : T-^oo) —> C be a positive definite function on J-coloured pair partitions, and H a Hilbert space. We define the Fock-like space ^jVa ®s (g) W® n(a)

•ft(W) := 0 n

'

(21)

a€l

with the factor ^y := Yla ^yr refering to the inner product. The symbol <S>S is a short notation for the subspace consisting of vectors ij} which lie in the range of the projection Pn U(T)®U(r).

(22)

r€S(n)

The operators U(T) act on the space <^)oej 7{®n(a) ^y permuting the vectors in each term On }~t(H) we define creation and annihilation operators for each index a & I and vector / € H. Let r&(/) be the creation operator which acts on the tensor product a}+s

"-b

(23)

as identity on 7^®"(a) for a 7^ 6 and as rigfot creation operator on 7i®n(6) . The annihilation operator at(/) acts on a vector of the level n of the Fock space as follows:

(24) As a consequence of 17 we have (25)

128

which means that a&(/) is a well defined operator on the dense domain •^t (%} °f tne Fock space .Ft (W) consisting of "finite number of particles" states. The creation operator acts on a simple vector vn s f := Pnwn ® f as (26)

We give without proof the following result which is a straightforward extension of the similar one in the case of one generalised Brownian motion 12. Theorem 2.1. Let /!,...,/„ be vectors in a Hilbert space "H. Then the expectation values with respect to the vacuum state pt of the monomials in creation and annihilation operators have the expression l

,-) (27) (v,c)€7>J(n)

(»,j)€V

where f Pt(abab) pt(abal)\

/01

Furthermore consider H = KC with K. a real Hilbert space and define the fields ujb(f) := «;,(/) + a^(/) for vectors / in K. Then the restriction of pt to the * -algebra generated by the fields is the gaussian state characterising the 2"-indexed generalised Brownian motion. Corollary 2.1. Let fi,...,fn

be vectors in the real Hilbert space 1C. Then

Remark 2.1. The restriction of the state pt to the algebra generated by the fields uJb(f) for a fixed b e X is a generalised Brownian motion in the usual sense. For different indices b one obtains in general different Brownian motions. An important class of positive definite functions on pair partitions are those which have the multiplicativity property t(V) = t(Vi)-t(V 2 )

(29)

for any pair partition V which is the reunion of two subpartitions Vi and V>2 not crossing each other. For a multiplicative function t, the operators ja are isometric (see lemma 5.9 in 12). Furthermore by proposition 5.10 of 12 , the field operators u>b(f) are in this case essentially selfadjoint.

129

3. The q-product of generalised Brownian motions Definition 3.1. Let (V,c) € Pj(oo) be a Z-coloured pair partition. The number of inter-crossings of (V, c) is denned by the total number of crossings between the pairs of different colours in V cr(V,c) =

{(P.?) I P>1 e V crossing, c(p) -£ c(q}}.

(30)

The main result of this work is the existence of the ^-product of generalised Brownian motions which satisfy the multiplicativty property. Theorem 3.1. Let 1 be an index set and — 1 < q < I. Let ta be a given multiplicative positive definite function on PI(OO) for every a € 1. Then the function (*11 t 0 )((V,c)) := H taCc-^a)).

(31)

is positive definite on Proof. We will firstly prove that for each n : I —» N the kernel fcn defined on the set of diagrams in BP^(a., 0) is positive definite. Let dj = (Va,i, /„ j)aei for i = 1,2, be two such diagrams with legs only to the left. Then we have

The kernel fcn can be written as a product of three kernels Mdi.da) = TT Mtdl ' d 2)a)' g cr ( dl ) +cr ( d2 > .q™(d^.

(33)

where by (d^ • d 2 ) Q we denote the a-colured component of dj • d 2 . The first product is a positive kernel by the positivity of each of the functions t a . The second product is also a positive definite kernel. The exponent cr(dj) stands for the number of inter-crossings of dj, that is number of crossings between pairs p, q with different indices - i.e., p e Va,i and q € Vb,i with a ^ b - plus the number of crossings between pairs and left legs with different indices - i.e., p = ( l , r ) € Va,i and k G La,i = Dom(/6W) such that I < k < r and a ^ 6. To obtain the total number of inter-crossings cr(dj • d 2 ) of the J-indexed pair partition d£ • d 2 , we need to add the number of crossings of right legs of d| and left legs of d2 which have different colours. This is the exponent cr(di,d 2 ) of the last term of the product 33. The factor qcr(~d^d^ depends only on the functions f^j which determine the positions of the left legs, and does not depend on the pair partitions Va,i- The positivity

130

of cr(di, d2) is thus equivalent to that of the vacuum representation of an algebra with commutation relations which is described below. Consider the algebra generated by the operators a\,ti with i = 1, ... ,n(6) satisfying the commutation relations a

b,iac,j - 1a,b a*,jab,i = $a,b8i,j

(34)

with qa,b = 1 if a = 6 and qa^ = q if a =£ b. Such algebras have been investigated in 13'14'25 and more generally in 7. There it is proved that for \q\ < 1 the algebra can be represented on a Hilbert space with vacuum vector Q satisfying a&^fi = 0. In particular this implies that the third kernel in 33 is positive definite and thus fcn as well. We denote the Hilbert spaces generated by the kernels kn by Vn. Let Xn:BPJ(n,Q)^Vn

(35)

be the Gelfand map, i.e. (A n (di),A n (d 2 )> = kn(di, d 2 ). On BPJ(n,0) there is an action of the group S(n) and kn is invariant under this action, thus it gives rise to the unitary representation Un on Vn. On the Hilbert space V := ®n Vn we define the operators ja by JaA n (di) = A n+5a (d0,0 • di).

(36)

The multiplicative property of the function t a implies that fcn(da,o • di, da,0 • d 2 ) = fcn(di, d 2 )

(37)

which means that ja is a well defined isometry. Moreover ja satisfies the intertwining property 17. We have thus constructed a representation of the "-semigroup BP^oo) on the Hilbert space V, with respect to the positive functional t where t = *ogjta. D

As described in the previous section we define now the creation and annihilation operators aj((/) on the Fock space J^c,) t (H) for / £ H in an arbitrary Hilbert space H and b e J. Similarly, the fields are Definition 3.2. Let t be a multiplicative positive definite function on (Iindexed) pair partitions. We denote by Ft(/C) the von Neumann algebra on J-t(Kc) generated by the selfadjoint operators u>b(f) with / in a real Hilbert space /C and b £ I. If the state pt on Tt(/C) is tracial then we call the function t tracial. For a complex Hilbert space Ti. we denote the von

131

Neumann algebra on J-t(H) generated by all the fileds wj,(/) with f £ H

Graded tensor product for q — — 1. We take a closer look at the case q = — 1. Let F be the unitary operator Ff = —f on H. Then

' Tt(F) : X >-* Ft(F)XFt(F)*

(38)

is an order two "-automorphism of Ft(7^) which we call Zs-grading. The vacuum state pt is invariant under F t (-F). This makes (Tt(H),pt) a Zjgraded non-commutative probability space 22. Definition 3.3. 22 Let (A, ) be a Z2-graded probability space with grading 7. Two von Neumann subalgebras A\ and ^2 of .4 which are invariant under 7, are called graded independent if the gradedly commute, i.e. 9a aia2 = (_i)9»i9«2 f or an a. e ^ whjch satisfy 70* = (-l) 'a;, where <9a.j € {0,1} is the grading of Oj, and moreover 0(aiOi2) — 4>(ai)4>(a2} • If .4 = AI V-^2 then we call A the graded tensor product of AI and ^.2From the definition of the — 1-product function *Jjgj t 0 we can conclude that the creation and annihilation operators of different index anticommute, i.e. af 1 (/ 1 )a^(/ 2 ) = -a» 2 (/ 2 )a» 1 (/ 1 )

(39)

for b ^ c. This implies that the algebra F + ( 9 ) , (H) is the graded tensor a£l

a

product of the non-commutative probability spaces Fto (H) for a £l. Corollary 3.1. The q-product *ogjta of multiplicative positive definite functions ta is a positive definite multiplicative function on P^(oo) and interpolates between the graded tensor product (q—-l), reduced free product (q=0) and the tensor product (q=l)- If all ta are tracial then *^gjta is tracial. The g-product provides a method for obtaining new positive definite functions on pair partitions by taking the product of known ones and restricting to a subalgebra generated by the sums of creation operators a&(/) over the same vector. Let I be a finite index set and t a be positive multiplicative functions for each a 6 2". On FfM t (K-) we define the new creation op} ^2b&iab(f}- ^e restriction of the state *{,cjtb to •\j\z\ the algebra generated by a"(/) is a Fock state and the associated positive erators a*(f)

:=

132

definite function on pair partitions (*bei^b)^ has the following expression in terms of t a and q: V|

C-»)

(40)

The restriction of a g-product of n functions which are equal to t will be denoted by t*n . We denote by tq the positive definite function arising from the algebra of g-commutation relations:

(41) The following central limit theorem states that for any positive definite multiplicative t, the g-product t*n converges to tq as n goes to infinity. Theorem 3.2. (Central Limit) Let t be a positive definite multiplicative function on pair partitions. Then t*71 converges pointwise to tq when n —> oo. Proof. Let V be a pair partition. Then E

4Cr(C'V) II tCc-'W)-

c:n-,V

(42)

a€X

for n := {!,... ,n}. We consider n big enough such that n > |V|. From the sum we isolate the terms which give to each pair in V a different color, c(pi) ^ c(pz) for pi ^ P2- Any such term brings a contribution equal to (I)M gcr(V)

which in total giyes g cr(V)_

The rest of the terms can be grouped according to the partition of V in sub-partitions determined by the colour of the pairs V0 = (c~1(o)} for a = 1 . . . ,n. A fixed such partitioning of V has at most |V| — 1 sets and thus the number of possibilities of atributing one of the n colours to each set is smaller that n^"1. In the limit n —> oo the contribution of this group of terms in the sum 42 tends to zero. D

Example. Consider the free Brownian motion, i.e. t(V) = 0 if V is crossing and t(V) = 1 if V is non-crossing. Then the product of n such functions gives /i\|V|

t 9* " ( V ) = (n\ J

. g c r W - ) ) { c : V - > r i | c r ( c - 1 ( i ) ) = 0 , Vi£n}.

(43)

Remark. At the moment is it not clear to us in what extent the g-product defined here for generalised Brownian motions can be extended to other

133 algebras. For example let A, B be two algebras as considered in generators satisfying commutations relations of the type fcj

7

with (44)

for A and similarly for B. Then the ^-product could be defined by adding the commutation relations aj6^ = qb^cii for all generators
References 1. F. Bergeron, G. Labelle and P. Leroux, "Combinatorial Species and Tree-like Structures" Cambridge University Press 1998. 2. M. Bozejko and Guta, M., Functors of White Noise Associated to Characters of the Infinite Symmetric Group, preprint August 2001. 3. M. Bozejko, B. Kiimmerer and R. Speicher, q-Gaussian Processes: Noncommutative and Classical Aspects, Commun. Math. Phys. 185, (1997), 129154. 4. M. Bozejko and R. Speicher, Interpolations Between Bosonic and Fermionic Relations given by Generalized Brownian Motion, Math. Z. 222, (1996), 135160. 5. M. Bozejko and R. Speicher, An Example of a Generalized Brownian Motion, Commun. Math. Phys. 137, (1991), 519-531. 6. M. Bozejko and R. Speicher, An Example of a Generalized Brownian Motion II, in Accardi (ed.) "Quantum Probability and Related Topics VII", World Scientific, Singapore, 1992, 67-77. 7. M. Bozejko and R. Speicher, Completely Positive Maps on Coxeter Groups, Deformed Commutation Relations, and Operator Spaces, Math. Ann. 300, (1994), 97-120. 8. D. Fivel, Interpolation Between Fermi and Bose Statistics using Generalized Commutators, Phys. Rev. Lett. 65,(1990) 3361-3364. 9. U. Frisch and R. Bourret, Parastochastics, J. Math. Phys. 11, (1970) 364-390. 10. O.W. Greenberg, Particles with Small Violations of Fermi or Bose Statistics, Phys. Rev. D 43, (1991), 4111-4120. * 11. M. Gu(;a and H. Maassen, Symmetric Hilbert Spaces arising from Species of Structures, preprint math-ph/0007005, to appear in Math. Z.. 12. M. Gu^a and H. Maassen, Generalised Brownian Motion and Second Quantisation, preprint math-ph/0011028 to appear in J. Fund. Anal.. 13. J0rgensen, P.E.T., Schmitt, L.M. and Werner R.F., q-Canonical Commutation Relations and Stability of the Cuntz Algebra, Pac. J. Math., 165, (1994), 131-151.

134

14. J0rgensen, P.E.T., Schmitt, L.M. and Werner R.F., Positive Representations of General Wick Ordering Commutation Relations, J. Fund. Anal., 134, (1995), 33-99. 15. A. Joyal, Une Theorie Combinatoire des Series Formelles, Adv. Math. 42, (1981), 1-82. 16. A. Joyal, Foncteurs Analytiques et Especes de Structures, in "Combinatoire enumerative", Proc. Colloq., Montreal/Can. 1985, Lect. Notes Math. 1234, (1986), 126-159. 17. Kostler, C.: "Quanten-Markoff-Prozesse und Quanten-Brownsche Bewegungen", PhD thesis, Stuttgart 2000. 18. B. Kiimmerer, Quantum White Noise, in Heyer, Herbert (ed.) et al., "Infinite dimensional harmonic analysis", Bamberg: D. u. M. Graebner, 156-168, 1996. 19. B. Kiimmerer, Survey on a Theory of Non-commutative Markov Processes, in L.Accardi, W. von Waldeenfels (Eds.): "Quantum Probability and Applications III", Lect. Notes Math. 1303, Springer Verlag, Heidelberg (1988), 154-182. 20. Kiimmerer, B.: A Dilation Theory for Completely Positive Operators on W*-Algebras, Dissertation, Tubingen, 1982. 21. H. Maassen and H. van Leeuwen, A q-deformation of the Gaussian Distribution, J. Math. Phys. 36, (1995), 4743-4756. 22. Mingo, J. A. and Nica, A., Crossings of Set-partitions and Addition of Graded Independent Random Variables, International Journal of Mathematics 8 (1997), 645-664. 23. Parthasarathy, K.R.: "An Introduction to Quantum Stochastic Calculus", Birkhauser Verlag, Basel,1992. 24. B. Simon, "The P($)a Quantum Euclidian Field Theory", Princeton Univ.Press, 1974. 25. R. Speicher, Generalized Statistics of Macroscopic Fields, Lett. Math. Phys. 27 (1993), 97-104. 26. R. Speicher, On Universal Products, in "Free Probability theory", Fields Institute Communications, ed. D. Voiculescu, Providence: AMS 1997, 257266. 27. M. Takesaki, "Theory of Operator Algebras I", Springer Verlag, New York, 1979. 28. D. Voiculescu, K. Dykema and A. Nica, "Free Random Variables", Providence RI: AMS, 1992. 29. D. Zagier, Realizability of a Model in Infinite Statistics, Commun. Math. Phys. 147, (1992), 199-210.

MULTIPLICATIVE PROPERTIES OF DOUBLE STOCHASTIC PRODUCT INTEGRALS.

R L HUDSON Department of Computing and Mathematics, Nottingham Trent University, Burton Street, Nottingham, NG1 4BU E-mail: [email protected]

1. Introduction.

Notions of simple and double product integrals, which are formal power series in an indeterminate h whose coefficients are in the first case finite sums of iterated quantum stochastic integrals, and in the second are elements of the tensor product with itself of the space of such sums, were introduced in [HuPul] and [HuPu2] respectively. In fact, as pointed out in [Hudsl], the underlying calculus needed to construct such product integrals is intrinsic to a given associative algebra, and does not depend on any representation by quantum processes in Fock space. The theory of simple product integrals was developed at this abstract algebraic level in [Hudsl]; here we outline a similar development of double product integrals. Unlike the simple case, where all the action can be regarded as taking place in the universal enveloping algebra of the Lie algebra got by taking commutators in the underlying given associative algebra, it is convenient to introduce a larger universal algebra, corresponding in the quantum realisation to iterated integrals defined by arbutrary tensors and not just symmetric ones [HuPu3]. The main purpose of this paper is to study the multiplicative properties of double product integrals. The simple product integrals of [HuPul] and [Hudsl] obey a simple and intuitive multiplication rule suggested by the distributativity of multiplication over addition. This simple multiplication rule holds also for double product integrals only in the case when the underlying given associative algebra is commutative as was pointed out in the quantum context in [HuPu2]. In the noncommutative case the situation is found to be more complex and the set of double product integrals does not 135

136 close under multiplication. Instead, it is embedded in a larger set of "perturbation double product integrals" which has the structure of a groupoid, in that each element can be multiplied on the left to give another element only by a subset (dependent on the chosen element) of premultipliers, and similarly on the right only by a subset of post multipliers. We shall find it convenient to use a definition of double product integral as an iterated simple product integral in each iteration of which there is an initial or system algebra. The equivalence, in the quantum context, to the definition given in [HuPu2] is proved in [Huds2] and in the abstract algebraic context in [Huds3]. Throughout we distinguish forward and backward directed product integrals, which in general do not have coefficients belonging to the universal enveloping algebra or its tensor square, from symmetrised product integrals which do so. The groupoid structure found is similar in each case.

2. Preliminaries. Let dL be a finite dimensional associative algebra. You may think of dC as the algebra of basic Ito differentials in a quantum stochastic calculus but what follows is in fact independent of any such interpretation and the underlying field F may be arbitrary. Let T(d£) and <S(cLC) denote respectively the vector spaces of tensors over dC and its subspace of symmetric tensors. Then [Hudsl] T(dC) is a unital associative algebra and S(dC) its sub-unital associative algebra under the multiplication (al3)N= Here the summation is over all ordered pairs (A, B) of possible emepty subsets whose union is NN = {1,2,..., N}. For n = 0 , 1 , 2 , . . . , an denotes the nth homogeneous component of the tensor a = (ao,ai,a 2 ,...). The notation aj^, indicates that the homogeneous component a^\ is to be placed in the tensor product of the \A\ copies of dC in ®N dL labelled by elements of \A\. oifA,/3PB, is formed using the multiplication in dL in copies of dC labelled by elements of A P| B in ®NdC, so as to obtain an element of ®Nd£. The unital associative algebra S(d£) may be identified with the universal enveloping algebra U of the Lie algebra got by equipping dL with the commutator Lie bracket under the universal extension (which is in fact an isomorphism) of the Lie algebra homomorphism

djCBdL^ (0, dL, 0,0,...) e S(d£).

137

There is a unique linear map d, called the differential map, from T(d£) to T(dC)®dC, satisfying the properties

d(ST) = d(S)T + Sd(T) + d(5)d(T) for arbitrary dL e d£ and S,T € T(d£). The latter property is called the Leibniz-Ito formula. Here T(d£)®dC is regarded both as an associative algebra under the natural tensor product multiplication, and as a two-sided T(d£)-module under the actions got by linear extension of S(T <8> dL) = ST ® dL = (S ® dL)T.

In the case when c?£ is the algebra of basic Ito differentials we may identify T(d£) with the unital associative algebra of finite linear combinations of iterated stochastic integral processes by linear extension of the identification

Then d maps each element of T(d£) to its stochastic differential, d maps the sub-unital algebra S(d£) to the subalgebra and submodule S(dL)®dL. In the case of quantum stochastic calculus, if S(dC) is identified with U, the identification with iterated stochastic integral processes may be regarded as the universal extension of the Lie algebra homorphism which maps each dL in dC to the corresponding basic process J0<(<. dL(t). Generally, the action of d on any homogeneous product tensor is to detach the final component to obtain an element of dC.;

d (0, 0, 0, . . . , (dLi ® dL2 ® • • • ® dLN) , 0, . . .) ( 0 , 0 , 0 , . . . , dLi ® dL2 • • • dLpt-i, 0, . . .) ® dLN In what follows, given an associative algebra ,4, we denote by^4[[/i]] the algebra of formal power series in an indeterminate h. Thus A[[h]] consists of formal power series A[h] = ^^=o hN AN with coefficients AN in A, which are multiplied by the rule oo

oo

oo

N

£ hNAN £ hNBN = X>" £ N=0

JV=0

N=0

j=0

We denote by A)[[^]] the ideal in A[[h]] generated by h; equivalently ^4o[ consists of formal power series whose zero-order coefficient is zero. The following Lemmas will be useful.

138

Lemma 1. -4o[[/i]] is a group under the product A[h] * B[h] = A[h] + B[h] + A[h]B[h\. Proof. It is straightforward to prove that -4o[[/i]] is a semigroup under * with neutral element 0. To prove the existence of inverses let us consider the equation A[h]*B[h]=0 for the unknown B[h], given A[h}. By equating coefficients we obtain successively Ai+Bi= 0, A2 + AiB1+B2 = 0 N-l

AN +

Evidently these equations can be solved uniquly for B\, B2,...', the solution is N

BN = ^(-l)k fc=l

£ ni+n 2 H

AniAH2...Ank

l-nk=N

where the summation is over all ordered fc-tuples (ni, ri2, • • • , rife) of positive integers whose sum is TV. A similar argument shows that the equation B[h]*A[h]=Q for the unknown B[h] has the same solution; thus A[h] has a twosided inverse. D We denote the inverse of the element A[h] in the group (.Ao[[/i]],*) by A[h]. For the next Lemma we assume that A is unital. Lemma 2. The set Q[[h]] of elements of -A[[/i]] of the form 1,4 + Y^N=I hNAN is a multiplicative group. Proof. This follows from Lemma 1 together with the fact that the elements of Q[[h]] are precisely those of the form 1.4 + A[h] where A[h] 6 and for two such elements we have (1A + A\h}) (1A + B[h}} = (1A + A[h] * B[h]). D

139 Now consider the unital algebra T(d£)[[/i]]. We extend the differential map d to elements of T(rf£)[[/i]] by action on the coefficients. Thus the indeterminate h is treated as a constant. The Leibniz-Ito formula continues to hold in the form d(S[h]T[h]) = d(S[h])T[h] + S[h]d(T[h]) + d(S[h])d(T[h]). Here T(d£)[[h]]®d£ is identified with the algebra (T(d£] ® d£) [[h]] in the natural way, and regarded as a two-sided T(d£)[[/i]]-module by F[[/i]]-linear extension of the rule hnS(hmT ® dL) = hm+nST ®dL = (hmS ® dL)hnT. We denote by e the unital associative algebra homorphism from T(d£) to F which maps each tensor a = (ao, 0:1, ay,...) to its zero-rank component QO- £ extends to a homomorphism, denoted by the same symbol, from T(d£)[[/i]] to F[[/i]], by acting on coefficients. Its restriction to U = S(d£) is the counit of the universal enveloping algebra I/, that is, the universal extension of the trivial Lie algebra morphism which maps every element of d£ to 0 F . Lemma 3. Let elements S[h], T[h] ofT(d£)[[h]\ satisfy dS[h] = dT[h], e(S[h]) = e(T[h]). ThenS[/i]=T[/i]. Proof. By linearity of d and e we have d(S[h\—T[h\) — 0, £ (S^/i]) — T[h]) — 0. The first equation shows that the non-zero-rank homogeneous components of the coefficients of S[h] —T[h] vanish. The second shows that likewise the zero-rank components vanish. Hence S[h] — T[h].O We note that an arbitrary element of (T(dC) ® d£) is the differential dT[h] of an element of T(d£) which becomes unique, in view of Lemma 3, if the value of e(T) is specified. Indeed, such an element T[h] (with e(T[/i]) = 0) is found by forming tensor products with basis elements of d£ of homogeneous components of each rank. But it is not true that every element of (S(d£) (g> d£) is the differential of an element of S(d£) 3. Simple product integrals. Let dS[h] € (T(d£) (g) d£)0 [[h]}. We consider the "differential equation" dU[h] = U[h]dS[h], e(U[h}) = 1

(1)

for the unknown U[h] € T(d£)[[h]}. Theorem 1. Equation (1) has a unique solution. It is of the form U[h] = 1 + V[h] where V[h] & T(dC)Q[[h}].

140

Proof. Let dS[h] = £~=1 hNdSN where each dSN e T(dC) <8> d£. Equating coefficients in (1), we get an equivalent system of equations for the unknown coefficients of U[h]

dU0 = 0, e(U0) = 1,

N

dUN = YiUN-jdSj, e(UN) = 0, 3= 1

which successively determine the coefficients UN uniquely. The solution of the first of these is C/0 = 1 so that U[h] is of the stated form.D In the case when dS[h] is of form 1 -j-(dc) ®dL[h], where dL[h] G d£o[[^]]i (1) becomes dU[h] = U[h] ® dL[h], e(U[h}) = 1.

(2)

In this case the solution can be written explicitly; it is given by oo

U[h] = l+^hN(0, aNil, ajv.2, . . . aN,N, 0,0,.. .) N=l

where dLri ® dLr2 <8> • • • ® dLrk fc=Ar

and the sum is over all ordered fe-tuples (n\, n?, • • • , rife) of natural numbers whose sum is N. Since each a^,k is evidently a symmetric tensor, U[h] 6 S(d£)[[h}} = U[[h\] in this case. We denote it by U[h] = JJ(1 + dL[h}) and call it the simple product integral generated by dL[h]. Theorem 2. Let dL[h],dK[h] 6 d£o[[h]]. Then

JJ(1 + dL(h}) J](l + dK(h\) = JJ(1 + dL[h] + dK[h] + dL[h]dK[h]) (3) Proof. U[h] = J](l + dL[h]) and U'[h] = Y[(l + dK\h]) are the unique solutions of

dU[h] == U[h] ® dL[h], e(U[h]) = 1,

141

By the Leibniz-Ito formula we have d(U[h]U'[h]) = (dU[h])V[h] + U[h] (dV[h\) + (dU[h}) (dV[h\) = (U[h] <8> dL(h]) V[h] + U[h] (V[h\ dK[h}) + (U[h\ ® dL[h}) (V(h] ® dK[h]) = U[h]V[h] ® (dL[h] + dK[h] + dL[h]K[h]). Also, since e is multiplicative,

e(U[h}U'{h})=e(U[h}}e(U'{h}) = l. Thus E/[ft][/'[ft] is the unique solution of the differential equation dV[h] = V[h] ® (dL[h] + dK[h] + dL[h]K[h]) , e(V[h]) = I , that is U[h]U'[h] = n(l + dL[h] + dK[h] + dL[h]K[h}} as required.D Thus the map dL[h] i-» J}(1 + dL[h\) is an injective homorphism from the group (cLCo[[ft]],*) to the group of elements of ZY[[ft]] having coefficient of h° equal to 1. Now let there be given an associative algebra A which we refer to as the system algebra. Let there be given an element of (A <8> dC)Q \[h}} oo

oo

n

n

dX[h] = Y hNdXN = Y h N=l

N=l

where each XN € A®dC, XN,J £ A, (dLl,dL?, . . . , dLn) is a basis of dC and each Xj[h] € A) [ft]- We are going to construct elements of (A <8> T(dC)) [[ft]] as solutions of differential equations. We refer to A as a left system algebra; in the analogous theory in which we construct elements of (T(dC) ® A) [[ft]], A is called a right system algebra. Assume first that A is unital. The forward and backward evolution equations generated by dX[h],for the unknowns U [ft] and U [ft] in (A ® T(dC)) [[ft]], are (id .4 ® d) U [ft] =U [h]r (I

T(dc}

® dX[h]) ,(idA®s)U[h] = !A

(id A ®d}U(h]=r (l T(dc) ® dX[h}) U [ft], (id^ ® e) [7 [ft] = 1 ^

(4)

(5)

where T is the linear map from (T(d£) ®A®dC) [[ft]] to (A ® T(dC) ® dC) [[ft]] which acts on coefficients by appropriately permuting the components of product tensors and we regard (A <E> T(dC) ® dC) [[ft]] as a two-sided (A <8> T(dC)) [[ft]]-module in the natural way.

142

Theorem 3. Equations (4) and (5) have unique solutions. They are of the form U [h] = 1+ V [h] and = 1+ V [h] where V [h],V [h] e (A®T(dC))Q((h\\. Proof. Equating coefficients in (4), we get a system of differential equations for the coefficients of U [h]

), (idA ®e) (Ui) = 0 A, N

(id A ® d) UN = E UN-J r (l T(dc) ® d^j) , (id A ® £) (f/w ) = 0 A,

where now T acts directly from T(dC) ® A®dCto A® T(dC] ® rf£, which successively determine the coefficients UN uniquely. The solution of the first of these is Uo= 1 so that [7 [h] is of the stated form. The argument for U [h] is similar.D The solutions U [h] and U [h] can be obtained explicitly. They are n

oo

N

E n^w

=lji,h,...,JN=l 1=1

dLJ2 < g > . . .®dLJN , Q , Q , . . . ) N

N

n

\^

V^

Z^

2^

k

V^

Z^

TT xr

\.\. n,ii

, 0, 0 , . . . ,
E

N = ljl,J2,...,JN

N

= 'i '=1

, 0, 0, . . . , dZ^'1 ® dP'2 ® . . . ® dZX", 0, 0, . . .)

143 oo

n

JV

k

E^E E

JV=1

fc=lj l ,J2,...,j*

E

n--

= l r i + r 2 + ---+r f c =JV i=l

, 0 , 0 , . . . , dLJ1 0 dLJ2 0 ... ® dL jfc , 0,0,.. .)-

-^yv

<_.w

In these expressions the directed products Yii=i an<^ EL=i run fr°m left to right and right to left respectively as / increases. We call [/ [h] and JJ

[h] the forward and backward product integrals generated by

dX[h] and denote them by H (1 + dX[h]) and Yl (1 + dX[h]) respectively. Note that, if A is noncommutative, neither of them belongs to (A 0 S(d£)) [[h]}= (A 0 W) [[/i]]. To obtain a product integral generated by dX[h] which belongs to (A 0 U) [[h]} we may replace the directed products by symmetrised products, thus we define the symmetrised product integral f^

£

f[Xjl(h]

N=l j l , J 2 , . - . , J W = l (=1

, 0 , 0 , . . . , dLjl 0 dLh 0 ... 0 dUN , 0 , 0 , . . . ) N

n

k

"£ E

JV=1

fe=l

E

}i,h,...,jk = lri+r2 + — +

), 0,0,..., where the symmetrised product Y[i=i is defined by

This may also be expressed as N

n

E E

E

showing that H(l + dX[h]) belongs to (A 0 U) [[h]]. If 4 is commutative, so that 0(1 + dX[h]) =[[ (1 + dX[h]) =H (1 + dX [h] ) , then a modification of the proof of Theorem 2 shows thatalign

dX[h\) JJ(1 + dy [/i]) = JJ(1 + dX[/i] * dY[h})

(6)

144

where dX[h]*dY[h] = dX[h\+dY(h]+dX[h}dY[h]. In the noncommutative case (6) holds neither for the symmetrised nor the directed products. But because the elements f|(l + dX[h}), Yl(l + dY[h]) and the corresponding directed products are invertible (by Lemma 2) there exist unique elements and dY[h] H ( l + d X [ h ] ) of (A®U) ({h}} such that

(7)

Similarly, there exist unique elements JI^x™ (^ + ^Y[h]), dY[h] II dX[h}), Ud[xh] (l+dY[h\) and dY[h] that

(l+dX[h]) of (A ® T(dL}) ([h}}such

[(l+dX(h])l[dX[h](l dX[h] + dY(h\ + dX[h}dY[h})

and

dX[h] + dY[h] + dX[h}dY[h})

We call ridX^jC 1 + dY[h]), terparts Udx[h] (1 + dY[h]),

dY[h]

dY[h]

11(1 + dX[h}) and their directed coun-

fi (1 + dX[h]), Rd[xh] (1 + dY[h]) and

dY[h] II (1 + dX[h]) right and left perturbation product integrals. The latter are respectively the solutions of the differential equations (id A ® d) U[h] = JJ(1 + dX[h])U[h]T (1 T(dC) ® dY[h}) JJ(1 + dX[h]), (id A ® d) £7[/i] = ?7[/i] JJ(1 + dY[h])r (1T(d£) ® dX[h]) J](l (id ^ ® d) C/^] = H(l + dY(h])r (1 T(dc} (id ^ ® d) U[h] = ^[(l + dX[h})r (1 T(dc)

145

together with the initial condition (id A ® £) U[h] = 1 ^ in each case. Indeed, multiplying the solutuion U[h] of the first of these on the left by TT(1 + cLV[/i]) and differentiating using the Leibniz-Ito formula shows that TT(l + dJf [/i])C/[/i] satisfies the differential equation of which J^(l+rfX[/i]* dy-[/i]) is the unique solution, and similar arguments establish the remaining claims. Theorem 4. Let dX[h], dY[h] and dZ[h] be elements of

=dz[h] H(l + dX[h]*dY[h]). Corresponding identities hold for directed perturbation product integrals. Proof. By repeated application of (7) we have, on the one hand dX[h]) H dx[h](l + dY[h}) H d X [ h ] t f d Y [ h ] ( l + Z(h}) dX[h] * dY[h]) H dX[h].dYW(l + Z(h}) = JJ(1 + dX[h] * dY[h]) * dZ[h}} and similarly

+ dX[h] * dY[h]) * dZ[h}). Thus

dX[h]) H dx[h](l + dY[h] * dZ[h}) and the first equality of the theorem follows by group cancellation. The second is proved similarly by multiplying both sides on the right by H(l +

dZ[h])n Theorem 4 shows that the sets of left and right perturbation product integrals have natural groupoid structures. For example Y[ dx[h}{^- + cfy[/i]) can be multiplied on the right only by an element of the form Ft dX[fc]*dy[h](l + Z[h}}. It can be multiplied on the left by 0(1 + dX[h}} or more generally by Y[ dxl[h](^ + dX2[h]) where dXi[h] * dX2[h] = dX[h]. In the case when A is not unital we can still define decapitated forward, backward and symmetrised product integrals Yl(l + dX[h\), ]\(l + dX[h])

146 and 0(1 + dX[h}) by ommitting the nonexistent 1 in the expansions which determine the corresponding undecapitated product integrals. In the case when A is commutative all three decapitated products coincide, and instead of (6) we have + dX[h\) * Y[(l + dY[h}) = f](l + dX[h] * dY(h}). Using the existence of inverses for the group operation * in (A®U}Q {[h]], in the noncommutative case we can define decapitated right and left perturbation product integrals ridxih]^ +c^['1]) and dY[/i]ri(l +dX[h]) such that

=

Y[(l+dX[h]*dY[h])

= UdY[h](l +dX^ * E(l +dYW)-

(8)

Corresponding to Theorem 4, these satisfy

H(l+dX[h]*dY[h}).(10) with corresponding identities for the directed product integrals. 4. Double product integrals. Let there be given an element dr[h] e (dC®d£)Q [[h]]. The directed and symmetrised double product integrals "Q JJ(1 + d?"[/i]), J^JJ(1 + dr[h]) and H n(1+dr['1]) can be defined as the elements of (T(d£) ® T(d£)) [[h]], (T(dC) ® T(dL}} [[h]} and (U ® U) [[h]] respectively

(i) i

n (

147

Here, in the inner decapitated product integrals the first copy of the associative algebra dC is taken to be a left system algebra, as is indicated by the superscript (l).Thus the decapitated product integral is an element of (dC ® T(d£)) [{h}} in the first two cases and of (dC®U) [[h}\ -in the third case, and the outer product integrals, in which the right system algebra is taken to be T(dC) in the first two cases and U in the third case, are meaningful as elements of (T(d£) ® T(d£)) [[h]] and (U ®U) [[h]] respectively. Alternatively we can define

(2)

+ n (i + in which now, in the inner decapitated product integrals the second copy of dC is taken as a right system algebra, and in the outer product integrals T(dC) in the first two cases and U in the third case are taken to be the left system algebra. That these definitions are equivalent to eachother and to a third definition used in the case of quantum stochastic calculus in [HuPu2] is shown in [Huds2, HudsS]. The underlying intuition is that the equivalence is a continuous analogue of the equality of the iterated ordered discrete products Y[f=i

( II^i^j,* ) an^ IJfc=i ( II j^i 2 ^ ) (an<^ con"

sequently of the iterated symmetrised discrete products Yl ^Li (II ^=ixj,k) and Y[ fe=i (I! *jLixj,k)) which holds whenever the Xjtk have the property that Xjtk commutes with xy ^ whenever both j ^ j' and k ^ k'. Theorem 5. Suppose that dC is commutative. Then, for dr[h], ds[h] 6

where dr[h] * ds[h] = dr[h] + ds[h] + dr[h]ds[h]

and we use the natural multiplication in the tensor product of associative algebras dC®dC. Proof. We use the first of the definitions of the double products. Using (6) and the corresponding multiplication rule for decapitated product

148

integrals in the commutative case we have

(i)

df

i

(i) i

n (! + W)) n( + n ( .D

In the noncommutative case the simple multiplication rule of Theorem 5 fails. But, using Lemma 2, we may introduce right and left perturbation symmetrised double product integrals defined by the identities

Hl[(l+dr[h}*ds[h])

together with their directed equivalents. They satisfy groupoid product relations. Theorem 6. Let dr[h], ds[h], dt[h] <E (dC®dC}0 {[h]]. Then

nn ^^(i+dsi/iDnn ^H^a+^D = nn Corresponding identities hold for directed products. Proof. The proof is similar to that of Theorem 4. Thus, for example, to prove the first identity we multiply both sides on the left by J| Yl(l +dr[h]). Using the defining identities we obtain in both cases H H(l + dr[h] * ds[h] * t[h}) so that the identity follows by group cancellation. D Like their unperturbed counterparts, perturbation double product integrals can be expressed as iterated simple product integrals with initial algebra, but the simple product integrals are themselves of perturbation type. Let us note first that the decapitated perturbation simple product integrals fidrl/ijC1 + ds[h]) and d»[h]t[W(i + dr[h}) in which the first copy of dC. in d£,®dC is taken as a left system algebra are elements of (dC. ®U)Q [[h]]. Thus we may form H(l +U%lh](l+d8[h])) and 11(1 +<*•[/.] ft ™ (1 + dr[h])) m(U ®U] [[h]]. Next we may form their perturbations Tin (

149 and

fl <'>(i+*[fc]) n(l+^]n (1) (l + ^W)). Similar considerations hold for directed double products Theorem 7 We have

1+ nn -M*i1 +*w> = n n'lHiW ns,]((i) dr i *w n ri( + w) =n <'>(!+*[*]) ii( +"M n (

Similar decompositions hold for directed double products. Proof. Using the decomposition H IK1 +dr[h]) = and the multiplication rules (7) and (8) we have

TT ( _i])^(ii ++ ll< TT( 11

1

)fl \ + dr\h]} I- >' * TT^ J.J.dr[/i]V(1 + ds\h]} I tl ]} if /

whence the firstidentity of the theorem follows by left-multiplying by the inverse of Y[ 11(1 + dr[h]).The remaining identities are proved similarly. D References. [Hudsl] R L Hudson, Calculus in enveloping algebras, Nottingham Trent Preprint, to appear in Journal of the London Mathematical Society (2000). [Huds2] R L Hudson, Algebraic stochastic differential equations and a Fubini theorem for symmetrised double quantum stochastic product integrals, to appear in " Quantum Information III" , proceedings, Nagoya (2000). [HudsS] R L Hudson, Calculus in enveloping algebras II, in preparation. [HuPul] R L Hudson and S Pulmannova, Algebraic theory of product integrals in quantum stochastic calculus, Jour. Math. Phys. 41, 4967-4980 (2000). [HuPu2] R L Hudson and S Pulmannova, Symmetrised double quantum stochastic product integrals, Jour. Math. Phys. 41, 8249-8262 (2000). [HuPuS] R L Hudson and S Pulmannova, Chaotic expansion of elements of the universal enveloping algebra of a Lie algebra associated with a quantum stochastic calculus, Proc. London Math. Soc (3) 77 462-480 (1998).

MARKOVIANITY OF QUANTUM RANDOM FIELDS IN THE #(X) CASE *

VOLKMAR LIEBSCHER GSF — National Research Centre for Environment and Health, Institute of Biomathematics and Biometry, Ingolstddter Landstr. 1, D-85758 Neuherberg, Germany, E-mail: [email protected]

We present a notion of quantum Markov random field based on a concept of conditional independence replacing the (usual) requirement of conditional expectation onto a desired algebra by conditional expectation onto a subalgebra, adopted to the special case that all subalgebras of the filtration are type I factors. It is immediate to introduce the classical notions like pairwise, local, global and factorizing Markov properties [11]. These share the same relations as in the classical case except the Hammersley-Clifford theorem, which remains open in the quantum case.

1. Conditional Independence At the heart of the study of Markov random fields in classical probability there is the notion of conditional independence. One says that two random variables X,Y are independent conditionally under a third random variable Z (notation X Jl Y \ Z) if

P(x e A,Y e B|z) = p(x e A|z)P(y e B\Z)

(i)

for any two Borel sets A,B. Equivalently it holds that P(X € A\Y,Z) depends on Z only. Then one considers the following key properties [11] (Cl) (C2) (C3) (C4) (C5)

IfX-lLF|Ztheny_lLX|Z. If X _LL Y\Z and U = h(X) then U1L Y Z. IfXALY\ZandU = ti(X)thenXALY\(Z,U). IfXiLKZandX_LLW|(y,Z)thenXlL(W,K)|Z. IfXILy|(Z,f/)andXJLZ|(K,t/)thenXlL(r,Z)|J/.

•This work was partially supported by INTAS grants N 96-0698 and 99-00545 and DAAD-DST grant

151

152

The first four hold in general whereas the last one needs additional requirements. Having such a notion of conditional independence, one can realize different notions for Markovianity for random fields indexed by the vertices of a finite undirected graph. If the law of the random field has a positive density with respect to a product measure, all Markov properties are equivalent and the density is ^e~H for a nearest-neighbour potential H. The last result is named Hammersley-Clifford Theorem, but many proofs can be found in literature before and after the appearance of [10], see e.g. [9,16,1 1,17]. In Quantum Probability, we we would like to follow this plan. First, we should "algebraize" the above conditions. I.e., we have to replace random variables by (the generated) subalgebras J3, "S, C, T>. Then the properties read as follows

(CA1) (CA2) (CAS) (CA4) If JUL'BIC and AUDI'S C thenll_LL\C. (CAS) Yet, before we can establish these properties in the quantum case, we need to know how to replace equation (1) since in general, conditional expectations onto a subalgebra need not exist [15]. One way out may be to use the Accardi-Cecchini conditional expectation [ 1 ] instead, but this map is not projective any more. A substitute may be the conditional expectations onto its fixed point algebra. There is an intensive study of Markov properties at least for faithful normal states [7,6,8] in these terms. Typically, one requires a large fixed point set of the Accardi-Cecchini conditional expectation, which is related to the fact that in quantum situations the local algebras are usually factors such that this fixed point condition is equivalent to assuming that the conditional expectation related to 25, C maps A into the boundary algebra C. Applications to quantum statistical mechanics of such conditions, mostly related infinite products of (Mn, can already be found in [3,4,2,5]. The aim of these notes is to clarify the structure of the Markov properties assumed there. Thus, we want to study one notion for conditional independence sharing as many as possible properties with the classical notion and being applicable at least to type I factors and also to nonfaithful states. This program is successful. We are able to show almost all of the above mentioned results for classical Markov random fields in the quantum setting. Unfortunately, we are not able to prove the Hammerley-Clifford theorem unless the graph is a tree. Once more, we want to point out that the key idea is to replace the conditional expectation onto some subalgebra by a conditional expectation into some

153

subalgebra. So it seems, that the proper notion of Markovianity in a quantum probabilistic framework (with the mentioned restricitons on the type of the von Neumann algebras) is found. 2. Notations Let 21 be a von Neumann algebra, co be a normal state on 21. If 2? C 21 is a von Neumann subalgebra, let co 2? with the additional property cL(ab} — <E(a)b for all a € A, b £ 25. For any map £ on a von Neumann algebra A let Fix(£) = {a € %. : £(a) = a} be the set of fixed points. 3. Quantum Conditional Independence We start by investigating various concepts as substitute for classical conditional independence. The key is not to require the existence of a conditional expectation onto a two-point algebra but just of transition operators with properties analogous to conditional independence. Below, we use for inclusions 'B C J2 the AccardiCecchini conditional expectation £!£ %, which is a special transition operator from %. to 2? the definition via Tomita modular theory can be found in [1]. Theorem 3.1. Let A, 'B, C C 21 be commuting von Neumann subalgebra s, each being a type I factor, and (ft be a normal state on 21. Then the following conditions are equivalent. (TO1)

There exists a transition operator £ : J?2JCi—> ISC such that CO = to o £

(CE1)

There exists a von Neumann subalgebra 'D, *B C 2) C 2JC and a conditional expectation "E : ABC i—> 2) such that co = co o £. (TOr) There exists a transition operator *E : RIS C \—> AC such that co = co o £ (CEr) There exists a von Neumann subalgebra 'D, %. C 2? C AC, and a conditional expectation "E : A.'SC i—> 2? SHC/Z f/iaf CO = CO o "E. (CEm) There exists an abelian von Neumann subalgebra Co C C and a conditional expectation £ : A'SC \—> Co such that CO = CO o £. Further, there is also a conditional expectation £1 onto C\ = C^ n C wifA to = to o £]. von Neumann subalgebra C\ splits into C\ = CgCg w/iere

154

Cg are commuting von Neumann subalgebras such that -E(acc'b) = £(ac)£(c'Z>),

(a e J2,Z> £
is faithful then the following two conditions are equivalent to the above ones. (ACCE1) £^c ^fulfils 0 C Fi (ACCEr) ^c^c fulfils & C Fi (fact) TTze density matrix p o/co on ABC fulfils p = pxcPvc sucn that p%c

Remark 3.1. We want to explain our reasoning behind the names of the conditions. (TO1) means that there exists a transition operator on the left of the triple (j?,C,!B). (CE1) or (ACCE1) mean that there exists a conditional expectation or Accardi-Cecchini conditional expectation projecting into the left of that triple. The meaning of (CEm) is the existence of a conditional expectation into the middle of the triple. Observe that (CEm) is a natural generalization of the classical conditional independence, especially equation (1), replacing the conditional expectation with respect to C by a conditional expectation with respect to an abelian subalgebra. Clearly, (fact) is a factorization property. The other names are straightforward modifications. Proof. It is clear that (CE1) implies (TO1). Conversely, assume that (TO1) is true. For a moment, we assume further that COc is faithful. Then it follows from the ergodic theorem that limw_>«, i £^=1 £" exists and it is the conditional expectation onto Fix(2;), which is a von Neumann algebra [1]. This shows the assertion in the case of a faithful to<;. If COc is not faithful, take the supporting projection P e C of COc [13]. Then co is faithful on P21P (which is again a type I factor) and there exists a conditional expectation TO : PACBP i—> PCBP such that the centre of Ca = !E(^C) and C\ = C^nC. Further, set C$ = C'A D C such that C& n C<s = CQ. Since every expected subalgebra of some ^B(y-T) (meaning that there is a normal conditional expectation onto it) is a type I algebra with discrete centre [14], we get C\ = C^Cg too. Further, this shows that there is a unique conditional expectation onto JIC\ "B, namely £1 (abc] = £; PiabcPi where the sum rums over the minimal (central) projections

155

of C\. Clearly, *£ o "Ei = *E such that we find ( O o < £ i = ( o o < E o £ i = c o o : £ = GD. We get for a £ A, b £ ), such that coCs3 = co o £'. It is easy to see that £' o £ leaves co invariant. Further, we derive for a e .£, fc £ ® and c € C^,c' € Cg £' o £(«c'd>) = £'(£(ac')£(c6)) = £' o <E(ac')<E' o <E(cb). This shows (CEm). Conversely, we want to infer from (CEm) on (CE1). We obtain from "E(acc'b) = rE('L(ac]c'b} that cOj?SCl is invariant under the map *£' = (El-flCfl) ®q, Id^ which is well-defined since for all minimal (central) projections P in C& fl-PC^P and 'BPC^P are type I and thus in tensor position. As a consequence, co is invariant under £' o £] which fulfils $ C Fix(£' o £]). Concerning the additional assertions for faithful states, it is obvious that (ACCE1) implies (TO1). On the other side, let £' : 2CBC i—> "SC be a transition operator into 'BC which leaves co invariant. Then by [Theorem 2.2][1], Fix('E') C Fix(£|!SC,BC). This shows $ C Fix(£™,gC sc) which is (ACCE1). Equivalence of (ACCE1) and (fact) was proved in [3]. The other implications are proved similarly. D Definition 3.1. Let (0 be a normal state on 21 and .#, 3), C three commuting von Neumann subalgebras. We say that SI and 'B are conditionally independent (under co) given C, symbol J^IL $|C, if condition (TO1) above is satisfied. Proposition 3.1. Suppose 21 is a von Neumann algebra with a normal state co and %., 'B, C, 'D are type I sub/actors. Then the following conclusions hold. (CQ1) (CQ2) (CQ3) (CQ4)

If A, 'B, C are commuting and AAL^C then c ^L and & _IL _LL are commuting and & 1L ®2>| C then & _LL S| C2). If A, $, C,
(CQ5) Suppose %., 'B, C, 'D are commuting and there are projections P% € A, P$ G "B, PC € C, P£, 6 2) iMc/z tfia? PfiP'sPcP'D is a supporting projection/or Wa'BC'b- If & -U- #| C2) an^f ^ _LL C| !B2) r/zen j? IL
156

Remark 3.2. We want to remark, that also in classical probability (CQ5) is not true in general, but only under requirements assuring that there are no trivial dependencies between the algebras. E.g., one uses the analogue of the above condition which says that the joint distribution has a positive density with respect to a product measure. Proof. (CQ1) is contained already in Theorem 3.1. (CQ2) follows immediately from restricting the transition operator "L in (TO1) to fD'BC. In the same way we see from ^1L #2)|C and (TO1) for the transition operator £ : R'BC'D \—> 'BC'D that Fix(£) 2 $£> 2 $ what implies AAL'B\C'D. Therefore, (CQ3) is shown. &JL2)\'BC, used with (TOr), provides us with a transition operator T,\ : R.'BC'D i—> R^C and -#JL <S\C used with the same condition with a transition operator £2: J^.'BCi—* Aft such thatFix(£i) I> .# C Fix( < E 2 )- Thus A C Fix(2;), £ = £2 ° £1 : AftC® i—> A® being a transition operator. This shows (CQ4). For proving (CQ5), observe that we can assume without loss of generality that MA'BC'D is faithful. Then we can use (ACCE1). Thus A _IL ®| C'D and A _LL C ®2> imply . This completes the proof. D Remark 3.3. (CQ3) is not a mere translation of (CA3), since (CA3) does not fit into our framework because J? and 2) C ft. do not commute in general. Nevertheless, our weaker version is strong enough to yield the important semigraphoid axioms in the next section. 4. Markov Properties on Undirected Graphs Let (V,E) be a simple undirected graph without loops and suppose that 21 is localised by ^P(V), i.e. for all A C V there is a type I factor 21^ C 21 such that 2U,2lfl commute whenever A n B = 0 and 2Uus = 21,421s as well as 2UnB = 2U A 21^. Let ~ denote the neighbourhood relation induced by E. I.e., x ~ y for *,;y e V iff (*,;y) 6 E. A c/tV/Me is a maximal set in V each element of which is a neighbour of each other element. Let Cliq(V,E) denote the set of cliques of (V,E). Now we consider a fixed normal state co on 21. We want to reformulate the conditions (CQ1)-(CQ5) from above, writing shortly A1L B|C if 2U -iL2lfl|2lc[a)]. (C'l) IfAlLB|CthenB-lL,4|C. (C'2) IfDcAandA_iLB|CthenD-lLB|C. (C'3) IfA_LLBUD|CthenAJlB|CUD.

157

(C'4) If A AL B\C and A 1LD\B \JC\henAALBUD\C. IfAlLB|CU£>andA_LLC|BUDthenA-LLBUC|D. These are just the gmphoid axioms of [12]. Without requiring (C'5), which needs additional properties of CO, the graph is called semigmphoid. The following conditions are the straightforward analogues of the Markov properties used in classical probability and statistics. (P) It holds {x} AL {y} \V \ {x,y} whenever x ^ y. (L) It holds {x} IL V \ {y : y ~ x or y = x} \ {y : y ~ x} for all x e V. (G) It holds A ALB\S whenever S separates A and B, i.e. there is no path from A to B which does not meet 5. (F) For the density matrix p of (f><&v it holds p = riceciiq(v,£) PC. where pc G 2lc are commuting positive operators. Remark 4.1. (P) is called pairwise, (L) is called local and (G) is called global Markov property. If (F) is fulfilled, the state CO is called factorizing. Theorem 4.1. For all normal states co it holds

If, additionally, CO is faithful then (F)=* (G) ^ (L) <==* (P). Proof. The proof of the corresponding classical results for the relation between (G), (L) and (P) relies only on the semigraphoid axioms (C'1)-(C'4), and, for faithful co, additionally on (C'5), cf. e.g. [12,11]. Therefore, we need to prove (F)=*(G) only. Suppose S separates A and B. Then (C'2) shows that we may assume without loss of generality that A and B are maximal, i.e. V — A U B U 5. Then every clique is either in A U S or B U S. Thus (F) shows that p = PAUSPBUS- This factorization D assures A JL B | S and completes the proof. Remark 4.2. We would like to have for faithful co that (F)«=(G). Unfortunately, we cannot show this result, which would be the quantum Hammersley-Clifford theorem, in general. On the other side, it is easy to see using condition (fact) in Theorem 3.1 that (P)=KF) it holds if (V,E) is a tree. Nevertheless, we have the strong feeling that this relation is true in general, but the proof requires a much deeper study of the expected subalgebras if the pairwise Markov property holds.

158

Acknowledgements Many thanks go to St. Lauritzen for his very demanding introduction to graphical models during SEMSTAT 1999. Further, we want to thank L.Accardi, F.Fidaleo and C.Cecchini for a lot of interesting discussions about conditional expectations, Markovianity and related topics. Without the kind hospitality during two stays at Centra Vito Volterra this work would not exist. References 1. L. Accardi and C. Cecchini Conditional expectations in von Neumann algebras and a theorem of Takesaki J. Fund. Anal. 45:245 - 273 1982 2. L. Accardi and F. Fidaleo Nonhomogeneous Markov states and quantum Markov fields Preprint Centra Interdipartimentale Vito Volterra Rome 462 2001 3. L. Accardi and A. Frigerio Markovian cocycles Proc.R.Ir.Acad. 83 A:251-263 1983 4. L. Accardi and V. Liebscher Markovian KMS-States for onedimensional spin chains Infinite Dimensional Analysis, Quantum Probability and Related Topics 2(4): 645-661 1999 5. L. Accardi and F. Fidaleo On the structure of quantum Markov fields submitted 2001 6. C. Cecchini Markovianity for states on von Neumann algebras In L. Accardi, editor, Quantum Probability & Related Topics VII World Scientific Publishing Co. Singapore 1992 pages 93-108 7. C. Cecchini A non-commutative Markov equivalence theorem In L. Accardi, editor, Quantum Probability & Related Topics VIII World Scientific Publishing Co. Singapore 1993 pages 109-118 8. C. Cecchini On the Structure of Quantum Markov Processes In L. Accardi, editor, Quantum Probability & Related Topics IX World Scientific Publishing Co. Singapore 1994 pages 149 - 157 9. R. Dobrushin Description of a random field by conditional probabilities and conditions of its regularity Theory Probab. Appl. 13:197 - 224 1968 10. J. Hammersley and P. Clifford Markov fields on finite graphs and lattices Unpublished manuscript 1971 11. S. L. Lauritzen Graphical models. Oxford Univ. Press Oxford 1998 12. J. Pearl and A. Paz Graphoids: A graph based logic for reasoning about relevancy relations In B. Boulay, D. Hogg, and L. Steel, editors, Advances in Artificial Intelligence volume II pages 357-363 North-Holland Amsterdam 1987 13. S. Sakai C* and W* Algebras Springer Berlin etc. 1971 14. E. Stoermer On projection maps of von Neumann algebras Math. Scandinav. 30:46-50 1972 15. M. Takesaki Conditional Expectations in von Neumann Algebras J. Funct. Anal. 9:306-321 1972

159

16. T.P.Speed A note on nearest-neighbour Gibbs measures and Markov probabilities Sankhya, Series A 41:184-197 1979 17. Winkler, Gerhard Image analysis, random fields and dynamic Monte Carlo methods: A mathematical introduction. Springer Berlin etc. 1995, 2nd edition 2003

ISOMETRIC COCYCLES RELATED TO BEAM SPLITTINGS

VOLKMAR LIEBSCHER GSF — National Research Centre for Environment and Health, Institute of Biomathematics and Biometry, Ingolstadter Landstr.l, D-85758 Neuherberg, Germany, J5-moi/:liebscherSgsf .de Abstract We provide in the context of quantum Markov chains due to ACCARDI coming up from iterated beam splittings a limit theorem concerning with a refinement of the discrete time. The limit object refers to a continuous time and should be called quantum Markov process. Closely related to our construction of such quantum Markov processes are isometric cocycles. We derive, upto technical conditions, a characterization of quasifree solutions of the cocycle equation.

1. Introduction In [1] there was introduced the concept of quantum Markov chains, closely related to finitely correlated states of [6,7,8]. [11] began investigations on quantum Markov chains related to so called beam splittings on symmetric Fock spaces. In [16] there was proven that such quantum Markov chains converge to quantum Markov processes with a continuous time parameter. In the present paper we find this limit theorem again, but in a different, more general formulation which is more convenient to deal with beam splittings connected with an inner system evolution. The transition operators related to the quantum Markov processes come from isometrics fulfilling the cocycle equation (19). This cocycle equation differs from the usual one (cf. e.g. [5]) by the fact that we do not consider unitary operators on a Boson Fock space but isometrics from a "smaller" Fock space. We characterize regular quasifree solutions of (19). General differential characterizations of such cocycles are subject to ongoing research.

2. Preliminaries For the natural numbers, integers, real, positive real and complex numbers respectively we use the symbols N = {1,2,...}, N0 = {0} U N, R, R+ =

161

162

[0, oo), C respectively. For a complex number z denote 3te resp. Qz the real resp. imaginary part of z. For any Hilbert space "H the symbol £(W) denotes the algebra of bounded linear operators on "H. If "H = L2(G, u) and / is a measurable function on G denote O/ the operator of multiplication with /. Any C* -algebra, A we consider will possess a unit denoted by I or I^t. A linear mapping P : A i—» B between two C"*-algebras A and B which is completely positive and unit preserving is called transition expectation. 2.1. The symmetric Fock space Let H be a separable Hilbert space, in the case of our interest H = L2(G, z/) where G is a Polish space and v a <7-finite Borel measure. The symmetric Fock space over "H is the Hilbert space

n=l

where Wj^, is the n fold symmetric tensor product of H [18]. An interesting class build the exponential vectors i^h £ r(W), /i £ Ti, defined through

As {VVt : /i G W} is linearly independent and total in F(W), we can define bounded operators on F(7i) by their restriction to the set of exponential vectors (cf. section 19 in [18]). Definition 2.1. Let HI, Hz be Hilbert spaces and C : HI \—> Hz be a contraction. Then the unique contraction F(C) : F(Wi) i—> r(W2) with F(C)V/ = Vc/ is called second quantization of C. If /i e n, the unique unitary operator W(h) £ £(F(ft)) with W(h)^f = e-MW-h+/

(1)

(2)

is called Weyl operator to the test function h. 2.2. Beam splittings and Quantum Markov Chains As ifrh (or better the normal state associated to this vector, the coherent state) represents some beam of bosons, we want to model the splitting of this beam. Consider operators 5, T € £(H) with 5*5 + T*T = I.

(3)

163

We define the operator VS,T • T(H.) \—> T(H) ® T(H) by extension of

It is easy to derive [9,10] that this operator is isometric and therefore

£$(A) = Vs,T*AVs,T

(4)

defines a normal transition expectation £*/£ : £(T(H) <8> Remark 2.1. In the case H = C which was exclusively handled in [3], S and T are just multiplications by complex numbers. As this case refers to a 1-mode boson field, these operators describe some reduction of intensity (the absolute value is the rate) and change of phase (the exponential of the argument times i). But also more general cases are thinkable. The case of multiplication operators on some L?(G, v) will be of special interest in the sequel. Then we think of the absolute value again as intensity reduction rate. In general, VS,T should be suitable to describe some phenomena where both splitting and scattering occurs. Remark 2.2. In case that both S = Oa and T = Op are multiplication operators on H = L 2 (G, v) we can rewrite (3) as a(x)|2 + |/?(x)|2 = l

jy-a.e.

(5)

This particular case was in the focus of [16], one basic of our considerations. The map £*/£ turns via the map w i-» w(£gp£) any normal state on £,(H) into one on £(W) <8>£(?f). We may apply this operation to one of the factors again and obtain a normal state on £(W) ® £(W) <8> £(W). Iterating this procedure more often we obtain a quantum Markov chain in the sense of [1,2], cf. [10] for an interpretation in terms of a measurement process. Definition 2.2. Let B be a von Neumann algebra and the quasilocal algebra S®N be the infinite von Neumann tensor product of identical copies of B. Furthermore, let £ : B®2 \—> B be a normal transition expectation and 77 be a normal state on B. Then the state w characterized by

• ® £(An ® I) • • •)))

(6)

for AI, . . . , An 6 B is called £ -quantum Markov chain with initial state rj. Remark 2.3. Since £ is normal u is a so-called locally normal state.

164

We are interested in the case B = £(F(W)) and £ = £*£% only. Then

w(Ai ® A2 ® • • • ®An I • • •) = »?(V5ir(Ai ® V^T(A2 ® • • • ® V|)T(An ® l)Vs,r • • •) VS.T) VS.T)

and we find for ,4 e S®" w(A ® I ® • • •) = r/(£*"04)) = r,(KAVn),

(7)

Vn = (I®" ® Vs,r) • • • (I ® VS,T) VS,T

(8)

where

n

defines another isometry and the definition of £* is obvious. The latter formula is also derived in the iterative procedure with Vi = VS,T and

Vn+m = (l®Vn)Vm.

(9)

Remark 2.4. The map V i—> I ® V is a shift operation. For continuous time, the analogue of (9) will be a cocycle equation with an analogous continuous shift, cf. (19). In the sequel we will incorporate besides the beam splitting an additional unitary evolution on F(7i) given by a unitary operator U but restrict S, T to act by multiplication. Then the transition expectation £ is defined through

with a, /3 fulfilling (5) and is again isometric. The operators V"'@'u are now determined by the iteration scheme (9) with V" = (I ® U)Voa,og3. Convergence to Continuous Time Quantum Markov Processes For practical purposes it is interesting to introduce also the continuous time analogue of quantum Markov chain. E.g. continuous beam splittings are of interest in models for telecommunication proceses, cf. [12]. The case of beam splittings driven by multiplication operators was already handled in [16], where we used the particle picture to show some convergence on the level of states or pointwise convergence of the kernels of their density matrices. To deal with the case where this pure beam splitting is accompanied by a certain unitary evolution (usually a free evolution) we want to go back to the level of operators. Our attempt is to construct isometric transition expectations (£**)teR with £**( • ) = V*( • )Vt where we get Vt via a limit

165

theorem from the operators Vn corresponding to multiple beam splittings, cf. (7), by making the time unit smaller and smaller. The finite time version of the discrete model lives on £(F(7i))®™. But £(r(W))® n is canonically isomorphic to £(r(H©W®.. .®H)) with n direct summands being identical copies of H. In the case of interest ®"=1 L2(G, v) is isomorphic to

This space may be isometrically embedded into L2(G x [0,1], v ® t |[ 0j ij) as functions constant on intervals (^, ^^]. The second quantization of the respective isometric mappings (the operator C\ below) is our main tool for putting all discrete settings into a continuous framework. More generally, we do not work on [0,1] alone but also on [0, t]. Then we can prove in certain cases strong convergence of the embeddings of the operators Vn coming from the (discrete time) quantum Markov chains described above to operators Vt mapping T(L2(G, v)) into T(L2((G x [Q,t], v®t)) ®T(L2(G, v)). Like in the discrete case we can introduce afterwards transition expectations £** by £*«( • ) = Vt*( • )V t . Algebraically we get the following framework for (£**) t€ jj • Let A be the C"*-algebra in £(T(L2(G x R+, v ® i |R + ))) generated by the local (von Neumann) algebras A

**HGx fo t) —

P^TV T 2 i/~i ^/ fn +\ j/ K?\ o \ fO tl \\\ /o» IF V \ \ I * /) ^ / / / ^ fi

oo")

+ r~^- TU il^4-.

In the usual manner there is the algebra ^Gx[ s ,t) such that for disjoint intervals the respective C*-subalgebras commute. There acts also the shift semigroup (s t ) teK+ on A, st transforming AGX[U,V) isomorphically into •AGx[u+t,v+t) for t > 0. By the limit theorems mentioned above we construct examples for the following definition. Definition 3.1. Let £*r : -4Gx[o,r) ® £(r(7i)) i—> £(r(7i)) be a normal transition expectation for all r £ R + . Assume (£*r)reis fulfils

5*S(A ® f*r(5 ® C)) = £*(r+s\Asr(B) ® C)

(10)

for all r,s>0,A& -4Gx[0,r), 5 e ^Gx[o,a) and C e £(F(H)). The (homogeneous) £ quantum Markov process with initial distribution 77 (which is a normal state on £(F(W))) is the state w on A determined by (A® I)),

r >0,Ae.4 G x [ o > r ) .

(11)

166

Remark 3.1. Clearly, (10) assures that w is well-defined, i.e. u(A) for a local operator A does not depend on the r > 0 for which A e ,/4.Gx[o,r)Again, a; is a locally normal state. In the case of simple beam splittings driven by multiplication operators it appeared [16] that u is even a normal state on T(L2(G x R+, i/ ® t |R+ )). This need not be true in the present, more general setting. The aim of the present paper is to investigate the structure of the limit of the operators Vn = V""''3'"^'1 in the following cases for the system evolution: • The system evolution is free, i.e. Un is a second quantization of some unitary (the free case). • The system evolution comes from Weyl operators (the WEYL case). • The system evolution comes from generalized Weyl operators as defined in [18] (the generalized WEYL case). This case contains both the first and the second one.

3.1. Preparation The following lemma will prove useful in the sequel. Lemma 3.1. Assume there are two Hilbert spaces Ti.^'H' and a sequence of operators (K) n€N , ||Ki|| < 1 from T(H) into T(H') fulfilling (VI) There are a dense domain Tio C H, mappings jn : "Ho '—> C and mappings Rn : "Ho '—> W such that (12)

for all h e Ho, n £ N. (V2) For all he Ho it holds

where 7 : W0 >—> C and R : H0 i—> H' . Then there is an operator V : T(H) i—> T(H'), \\V\\ < 1 with

and V fulfils

167

Proof. As {4>h : h € Ho} is total in T(H) and all Vn are contractions it is enough to prove

for h € Ho- Continuity of the map h i-> ^ completes the proof.

D

In the following we will use the operators C* : T(L2(G, v))®n \—> r(L 2 (Gx [0, t),v®i |[0,t) )) determined by C£(tfhl<8>---®V;,J = ^

(13)

with h(x,s) = Being the second quantization of the isometry from ®"=1 £ 2 (G, v) into L 2 (G x [0, t},v®l | [0)t) ) given by /— n hi ® h2 ® • • • © hn i-» J^ V xri=i n t ni t) ( • }hi

C^ is an isometry. Remark 3.2. We will use the operators C^ to embed discrete tensor products into continuous ones. This idea is useful to translate models on different Hilbert spaces into one universal model, cf. [15] for another application. A similar scheme was used in [17].

3.2. The Free Case In this section we will consider a free evolution of the system, i.e. U = T(V) is the second quantization of a one particle unitary V. We know the action of Vn on exponential vectors:

fc=l We will embed these operators in the same Fock space by use of (C£ ® I). Consider the operators (C£ ® I) o V™"1/3"'r' "' with sequences (o;n)neN, (/?n)rjGN and (Ki) neN . To obtain a nontrivial limit as n —> oo we need a suitable behaviour of (a n ) n6N , (/3n)n6N and (Vn)n£N.

168

So V will be related to a Schrodinger evolution in a time interval of length ^ w.r.t. a one particle Hamiltonian H (which is selfadjoint), i.e. (14)

The function f3n will be given by /^(x)=e-^'),

(15)

where r : G i—> C fulfils 3£r > 0 z/-a.e. This can be interpreted as shrinking also the time interval for the splitting. With regard to (5), we set

M*) = | v i - | / W I 2

(is)

where p is some function with |p(z)|2 = 25ftr(:r) and § :— o. To get convergence of the operators (C* <8> I) o v"n'/3"'r(V'n) the limit

shall exist (it should define a strongly continuous contraction semigroup then). Due to the Trotter formula [4],Corollary 3.1.31, if Ut = e~im and H — iOr exists densely, is closable and its closure H ' generates a (contraction) semigroup, we may conclude that

uniformly for t in compacts. Moreover, as both —iH and — Or are dissipative (i.e. 3?{/, -Ltf/) < 0 for all /, cf. Proposition 3.1.15 in [4]), also — iH — Or is dissipative on dom(H) n dom(Or). Thus (cf. loc.cit.) iH — Or is closable. Prom this we get sufficient conditions for convergence to a semigroup. Unfortunately, this is not enough for our purpose, as the term for an has not yet been considered. To overcome this difficulty we need further conditions, we take a rough but simple one. If both p and r are bounded, any operator of interest is defined on dom(H ), which makes all calculations very simple. E.g., — iH — Or is closed on dom(.ff) and generates a strongly continuous semigroup Proposition 3.1. Assume r and p are bounded. Then for all t > 0

(C*®I)oV^">™ —->Vt n—>oo

where Vtiph = iph' <8> tpwth with h'(x, s) =

X(o,t)(s)p(x}(wth)(x).

169

Proof. In the spirit of (12) there correspond to V"" and an operator Rn given by

a function jn

Rn(h) = k=l

for h € dom(H). After application of C£ I we get

fc=i By Corollary 3.1.31 from [4] the terra (e '™ f f O ^-^ /i converges as n —> oo to Wsh uniformly for s € [0,t\. Thus

fc=i converges uniformly to /i'(s) in L2(G, v) for s € [0,i]. Now we observe

and | Y^jan(a;)| < b( a: )l- This implies that even

fc=l

uniformly for s e [0, £]. Consequently,

and Lemma 3.1 completes the proof.

D

Remark 3.3. Similar statements hold if (V t ) teR is only a contraction semigroup. To define the operator Vt, the boundedness condition on r can be relaxed considerably (see section 4 below).

170

3.3. The Weyl Case Now we assume U — 2U(/), a short calculation shows for all f,h£ L 2 (G, v

k=l A strongly continuous one parameter group consisting of Weyl operators has the form Ut = W(tf). As before we use in the discretization the time step £ for the unitary time evolution. The parameters an,(3n are built from functions p, r as in the preceding section. We use the conventions

l

1

Proposition 3.2. If f £ dom(Op) then (C* B ®I)oV^ A '- H r ( i / ) -=— V, n—»c» w/iere I t e-'^ + tr-l

'

Proof.

^

f\

f

/ j ll-e-'l -e-'lhX

~f'^—

(17)

We rely on Lemma 3.1. Similar to the free case we obtain

Elementary calculus gives

t :f o

/ -,

^^ if/3 = l

171

This implies lim - y^(/3n)l(z) = < "^°°n fct [ ^^ '"

. -.

I

tr x

() 1

if r(z) = 0 JL

II f \ X } — U

where the convergence is uniform on sets where • 9?r is bounded away from 0 or • 5Rr = 0 and er is uniformly separated from 1 or • r is equal to 0. Additionally we know that jan(x) L

pointwise, but uniformly for 9£r in bounded sets. Consequently, both In 7^ (ft) and R'n(h) converge on a dense domain completing the proof. D

3.4. The Case of Generalized Weyl Operators Let V e £(L 2 (G,z/)) be unitary, / 6 L2(G,v). According to [18], generalized Weyl operators are given by

and generalize both second quantizations and ordinary Weyl operators. Moreover, W(f, V) = W(f)T(V). Thus we need no additional rules of calculus. Vh = Jn(h)i^Rn(h) with Similar to the Weyl case we obtain V"

-In 7n (ft) =

||/||2 + {/,£(n

k=l

A one parameter group of generalized Weyl operators Ut = W(ft,Vt) is given by a unitary group (V r t ) teR , Vt = e~ttH , and (/t) teR which fulfils f. + V.ft = f.+t,

s,t>0.

172

Solutions of this equation will be discussed in greater detail around Lemma 4.3 below. We just remark that if 11-> ft is differentiable there is a vector / € L 2 (G,i/) for which t t ft = j t(ds)VJ = I e(ds)e-isfl f. (18) o o For the following statement we omit the proof as it follows the same lines as the Propositions 3.1 and 3.2. Proposition 3.3. Assume that r is bounded and (/t) teR fulfils (18) for some / £ L 2 (G,i/). Then (C* ® I) ov ->^(A/^/~) __!__ Vt n—>oo

where Vt is given in the spirit of Lemma 3.1 by the following maps *yt and

Rt. -ln( 7t (/0) = (/, fl(ds)(t - s)WJ) + (/, JO

„(•)

ft(d8)W,h) JO

Rt(h) = X(o,t](' )(0 P / ^(ds)Ws/ + OpW(.)h) Jo rt ®( £(ds)W.f + Wth)n Jo 4. On the Shift Cocycle Property We are interested how general the examples for operators Vt obtained by the limit theorems in the above section are. From the discrete time formula (8) we get a continuous time analogue which reads as

o vs = vs+t

(19)

for all s,t e M+ where Vt : M t—> jM(G x [0,t),v x ^|[ 0 ,t)) ® A^ is an isometry for all t > 0. Thereby Bs : L2(G x R+) 0 L 2 (G, i/) i—> L 2 (G x [s,oo)) ® L 2 (G,ix) is the right shift by s in L 2 (G x K+), cf. (25). As -^XfGxfo si i/x<| ) ®r(0 s )V t represents also a shift operation, we can think of (Vt) teR as shift cocycle. In the sequel we will search for solutions of (19). From this equation it follows that the liftings £**( • ) = V*( • )Vt will automatically fulfil (10). Equation (19) seems too general to obtain a complete explicit solution. We will only search for solutions based on an ansatz near to generalized Weyl operators: Assume Vt is given by (20)

173

with7t : Ho K-> C, Rt : H0 H—> L 2 (G x [0,0," ® * |[o,t)) © L2(G,v). In the sequel, we use the set <£f(Ho) = span {iph : h G HO}. First we solve the problem under which conditions operators Vt fulfilling (20) can be isometries. Proposition 4.1. Let H, H1 be two Hilbert spaces and V : T(H) i—> T(H') be an isometry which can be given on
where 7 : Ho i—> C and R : Ho '—> H' are arbitrary (i. e. may be nonlinear) maps. Then R is an affine map. More exactly, there exists an isometry VQ : Ho i—> H1, v0 & H' and c e [0, 2n) such that

R(h) =v0 + V0h •j(h) == e ic e-5ll"o|| 2 -i> Proof. First we introduce the concept of analytical maps on Hilbert spaces. For a map S : H \—> H' where H is a vector space and H' is a Hilbert space we say that S is strongly analytic if for all hi, . . . , hn € H the map (21, . . . , zn) i-» S(zih\ + • • • + znhn) is an analytic H' valued function on C™. It is well-known (cf. Proposition 20.2 in [18]) that the map h i-> Vfc is strongly analytic. Moreover, strong analyticity is preserved under bounded linear maps such that we can conclude that "Ho 3 h i-> 7(/i)V>fl(h) is strongly analytic. Projecting on the vacuum shows analyticity of 7. As V is isometric, f ( h ) ^ 0 for all h € Ho- Thus, h H-> -—• is analytic too and we derive from projecting on the first Fock layer that R itself is strongly analytic. Now take any ho, hi, h'0, h( e Ho and set h = h0+zhi and h' = h'0+z'h'i. As V is an isometry we get

As this is non-zero there is an logarithm of 7 which constitutes again an analytic function (cf. [14]). So we take the logarithm of the above equation: In j(h') + In 7 (/i) + (R(h'), R(h)) = (h1, h).

(21)

Now assume h = h0 + zhi, h' = h'0 + z'h( and expand the left hand side into power series. It should hold

n=0

174

n=Q

R(h0+zh1) = n=0

n=0

with K) n € N ,(^ n ) n e N C C, (r n ) n6N , (r' n ) n6N C L2(G2,V2). From equation (21) we derive that all mixed terms of order greater than 2 in the expansion of (R(h'),R(h)} have to vanish. E.g. the coefficient of z 2 z' 2 is ( r 2> r 2) giving (r^r-i} — 0. As the choice of h0,hi, h'0,h( e Ho was arbitrary, set ho = h'0 and hi = h\ giving r'n = rn. So we derive r^ = 0 and in the same manner rn = 0 for n > 2. Thus R and In 7 are in fact affine. Inserting this into equation (21) we get

(h'0, ho) + z(h'0, hi) + z'{h(,h0) + zz'^hi) giving d'0 + d0 + (r'0,r0) = (ti0,h0)

(22a)

dr+(ri,r 0 } = {h' 1 ,ho> di + (r'0,n) = (ht>,hi)

(22b) (22c)

<»-i,ri) = {/il,fci>

(22d)

From definition it is clear that the correspondence hi H-> ri is linear. By the equation (22d) there is an isometry V0 with r\ = V0hi for all choices of hi. We write R(h) = VQ + Voh. The choice ho = h'0 yields (observe VQ — TO = TO) for arbitrary /ii = h[ 2ftd0 + \\v0\\2 = 0 di + (v0, V0hi] = 0. We see do = — ^\\vo\\2 and di = — (V^vo, hi) and arrive by setting Qd0 — c (mod 2vr) immediately at the assertion. D In our case we get

(23) where

175

Vt = ft © 9t Vth = Wth © Wth

with ft

eL2(Gx[0,t),i/x*|[0it))

In the following we embed £(L 2 (G,z;),L 2 (G x [0,t),i/ <8>^|[o,t))) into £(L 2 (G, i/), L 2 (G x R + , ^ ® £ |R+ )). As Vt must be isometric, we derive

w;wt + w;wt == iLa(0il/)

(24)

This means that (Wj) t6R are contractions. Denote by 9t the shift by t in L 2 (G x R + , v ® i |R+ ), (0t&)(a,*) = X[t,<x>)(*)>i(s-*,z).

(25)

Direct calculations give: Lemma 4.1. IfVt is an isometry given by (20) equation (19) is fulfilled if and only if ct+s = ct+cs

(mod 2?r)

(26a)

Jt+s = ft. + Otfs^+ etWsgt Wt+s =Wt + 6tWsWt

(26b) (26c)

9t+s =9t + Wtgs Wt+s = WtW.n

(26d) (26e)

For "good" solutions we need some (strong) continuity. Lemma 4.2. Vt is strongly continuous if and only if (1) t \—» elc* zs continuous. (2) 1 1-* ft and 1 1—> ^4 are continuous. (3) 1 1-> Wt and 1 1—> Wt c""e strongly continuous. Proof. It is clear that (Vt) t6R is strongly continuous if and only if t H-> VtV1/! is continuous for all h £ L 2 (G, i/). Then (1) follows from the continuity of 1 1—> (0o, Vt^o), (2) from continuity of i i—> VtV'o and (3) from the general case. The converse follows from continuity of the map h i-> -iphd

176

By continuity and (26a), (ct) t€K can be choosen as ct = ct for some c e R. Then Vt = e-ic*Vt fulfils again (19) and Vt*( • )Vt = Vt*( • )V t . Thus we will set c := 0 in the sequel. As (Wt) teR+ is a strongly continuous contraction semigroup its generator H is a well denned closed operator (in fact a so called maximal accretive operator, cf. [13]). This parametrizes (Wt) teK , the second question poses (<7t) t€ R . We have only partial answers to solutions of equation (26d). Lemma 4.3. Let (Wt}t£^ be a strongly continuous semigroup of contractions on L2(G,v) and (<7t) t6 R be a differentiable L2(G,v) valued function fulfilling (26d) for all s,t 6 R+. Then there is a vector h £ L2(G,v) such that gt = [* t(d8)W,h. Jo

(27)

Proof. Differentiating (26d) at s = 0 gives (set g'to = Wtg'0 = g't. By taking in (26d) the limit t J, 0 we get go = 0. Thus we derive with

9t = go + I t(ds)g't = I £(ds)Wshn Jo Jo

n

A further result concerns functions (<7t) t6 R which are continuous only. Lemma 4.4. Let (Wt)t 6 R be a strongly continuous semigroup of contractions on L2(G,v) and (fft) t€ R be a continuous L2(G, v) valued function fulfilling (26d) for all s,t £ R + . Further, assume that Wt — I is invertible for a dense set of t £ R + . Then there is a vector h £ L2(G, v) such that 9t

= (Wt - I)/i.

Proof. For arbitrary s, t we derive from (26d) gs + Wsgt = ga+t = gt or

(W, - %t = (Wt - !)
(Wt - I)-V = (W. - I)-^

(28)

177

Thus for a dense set of t G R+ gt = (Wt - l)h

which carries over by continuity to all t 6 R + .

O

Remark 4.1. Invertibility of Wt — I for a dense set of t means 0 g spec(H). We can use this lemma also for the case where (W t ) teR has invariant vectors: Denote g° the projection of gs onto the subspace of invariant vectors being Ker(H). Then Continuity gives g® = tg' for some g' g Ker(H). Moreover, if the semigroup (W t ) teK restricted to Ker(.H')1 fulfils the condition of the lemma above we may conclude gt = th' + (Wt - I)h

(29)

where h' is an invariant vector. Remark 4.2. Concerning Lemma 4.3 we remark the following: Suppose h g Ran(tf ) ® Ker(ff), i.e. h = Hh + h' with Hh' = 0. Then / l(ds)W,h = [ t(ds)Ws(H~h + ti) = I e(ds)W3Hh + f t(ds)h' Jo Jo Jo Jo = th' + Wth-h and we arrive at (29). We present one solution of (19) under technical assumptions. Proposition 4.2. Assume (Vt) t6K is a strongly continuous shift cocycle of isometries fulfilling (20). Then (M / t)t 6 R is a contraction semigroup with generator H. If H is bounded, H~l exists and is again bounded then (fft) t £R is given by (28). Furthermore, there are a bounded operator A g £(L 2 (G, */)) with A* A = H* + H, and a vector f g L 2 (G, i/) such that f t ( x , s) = X(o,«] (*)(/(*) + Ag,(x)) Wth(x,s) = X(o,t](s)(AW3h)(x),

(30a) 2

(h 6 L (G,^))

(30b)

Proof. We derive from equation (26c) for s,t g R + , h £ L 2 (G, i/) and ^-a.a. u < t Wt+lh(u) = Wth(u).

178

Thus there is an operator W : L2(G, v) \—> L2(G x R+, i> t |R+ ) with

for all t £ R+ and £-a.a. s < t. From equation (24) we get \\W\\ < 1. Now we consider, as t [ 0, for h £ L 2 (G, v)

(Due to our assumptions, I — Wt* is invertible for small i.) (26c) yields and

By continuity, we can set B/i = (I — W£)~lW*(h ® X[o,t]) for t small enough. This implies

and

Consequently,

and by continuity we derive i?*/i'(z)= f

e(ds)W*H*Bh'(x,s).

R+

Under the setting A = B*H we arrive at

which determines the structure of Wt. The equation A*>1 = H* +H follows from (24) by differentiating at t = 0. In the same manner as above we get from equation (26b) the existence of / e L 2 (G x R + , v ® ^ |R+ ) such that

ft(x,s)

=X[o,t}(s)f(x,s)

179 which yields

Thus i H-> /( • , t) is continuous, the limit s J. 0 gives

Setting /(a;) = /(a;, 0) we arrive at (30a) and the proof is over.

D

At the end, we present more general quasifree solutions of (19). The easy proof is omitted. Proposition 4.3. Assume (Wt)t&A is a contraction semigroup with generator H such thatH+H* is densely defined. Further, suppose that (gt) teK fulfils (26d) (e.g. it is of the form (29)). Take a closed operator A with A* A 2 H* + H such that gt 6 dom(A) for all t e R+. For a vector f € £ 2 (G, i>) we define

ft(x,s) = x[0,t) (*)(/(*) + Ags(x)} Wi/i(x,a) = X[o,t)(s) W/i)(*)

(32a) (32b)

ai least for h € dom(.H"*+.H"). Lei Vt 6e isometrics extending (20) and (23). Then (Vt) tgR is a strongly continuous shift cocycle, i.e. it fulfils (19). D Remark 4.3. We want to compare this to Proposition 3.3. Fix / e L 2 (C? ) i'), rate functions p, r and a self adjoint operator H. We see from Proposition 4.1 for /i £ dom(JJ) n dom(O r ) S

o

l(du)Wuf(x)

gt(x)= [* l(ds)(WJ)(x) Jo Wth(x,s)=X[0,t)(s)p(x)(W.h)(x) and H = \H + O r , i.e. Wt = e-ttH-to- . This is the same as (32a), (32b) if we set A = Op and / = 0 and (#t)(eR suitably. How can we interprete the additional / in (32a)? Actually, there is a simple solution to (19), namely Vt = 2U(/ <8> X[o,t]) ® I with arbitrary / € L2(G,v). We can interprete this as constant background creation of bosons in mode /. Surely, we can also give limit theorems for this more general type of operators by adding this background operators also in discrete time, e.g. by replacing Vn with

180

References 1. L. Accardi Noncommutative Markov chains In Proc. Int. School of Mathematical Physics, Camerino pages 268-295 1974 2. L. Accardi Topics in Quantum Probability Phys. Rep. 77:169-192 1981 3. L. Accardi and M. Ohya Compound Channels, Transition Expectations and Liftings Appl. Math. Optim. 39:33-59 1999 4. O. Bratteli and D. Robinson Operator Algebras and Quantum Statistical Mechanics 12nd ed. Springer Berlin Heidelberg New York 1987 5. F. Fagnola Characterization of Isometric and Unitary Weakly Differentiable Cocycles in Fock Space In L. Accardi, editor, Quantum Probability & Related Topics VIII World Scientific Publishing Co. Singapore 1993 pages 143-164 6. M. Fannes, B. Nachtergaele, and R. Werner Abundance of Translation Invariant Pure States on Quantum Spin Chains Lett. Math. Phys. 25:249-258 1992 7. M. Fannes, B. Nachtergaele, and R. Werner Finitely Correlated States on Quantum Spin Chains Commun. Math. Phys. 144:443-490 1992 8. M. Fannes, B. Nachtergaele, and R. Werner Finitely Correlated Pure States J. Fund. Anal. 120:511-534 1994 9. K.-H. Fichtner, W. Freudenberg, and V. Liebscher Time Evolution and Invariance of Boson Systems Given by Beam Splittings Infinite Dimensional Analysis, Quantum Probability and Related Topics 1(4):511-531 1998 10. K.-H. Fichtner, W. Freudenberg, and V. Liebscher Beam Splittings and Time Evolutions of Boson Systems Technical Report Math/Inf/96/39, Fakulty of Mathematics and Computer Science Jena, 1996 11. W. Freudenberg On a Class of Quantum Markov Chains on the Fock Space In L. Accardi, editor, Quantum Probability & Related Topics IX World Scientific Publishing Co. Singapore 1994 pages 215 - 237 12. J. R. Jeffers, N. Imoto, and R. Loudon Quantum Optics of Travelling-Wave Attenuators and Amplifiers Phys. Rev. A 47(4):3346-3359 1993 13. T. Kato Perturbation Theory for Linear Operators third edition Classics in Mathematics Springer Berlin Heidelberg New York 1995 14. L. Kaup and B. Kaup Holomorphic Functions of Several Variables de Gruyter Berlin New York 1983 15. V. Liebscher On a Central Limit Theorem for Monotone Noise Infinite Dimensional Analysis, Quantum Probability and Related Topics 2(1):155-167 1999 16. V. Liebscher A Limit Theorem for Quantum Markov Chains associated to Beam Splittings Open Sys. Inf. Dyn. 8:1-29 2001 17. K. Parthasarathy The Passage from Random Walk to Diffusion in Quantum Probability I J. Appl. Prob. special volume 25A: 151-166 1988 18. K. Parthasarathy An Introduction to Quantum Stochastic Calculus Birkhauser Basel, Boston, Berlin 1992

MULTIPLICATIVITY VIA A HAT TRICK

J MARTIN LINDSAY AND STEPHEN J WILLS School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD E-mail: [email protected], [email protected] Multiplicativity for Fock-adapted regular Markovian cocycles is proved for two cases, the first of which is new and facilitates dilation of quantum dynamical semigroups on a separable C* -algebra. The proof of multiplicativity uses complete boundedness of the stochastic generator and the cocycle, induced maps on matrix spaces, and a simple commutative diagram.

Introduction Let j be a Fock-adapted regular Markovian cocycle on A with noise dimension space k, where A is a unital (7*-algebra acting on a Hilbert space I), and k is a separable Hilbert space. If j is completely bounded then the cocycle relation it enjoys may be expressed simply is+t='!is°asojt

(1)

where a is the Fock space shift semigroup and j~s is the map between matrix spaces M(Jrs; A)^ and M(Jr; .4)^ (defined below) induced by js. Regularity for the cocycle means that its associated semigroups are norm continuous. If j is completely positive and contractive then it has a completely bounded stochastic generator in the following sense.8 There is a completely bounded operator 6 from A into the matrix space M(k; A)-^ (where k := C © k) such that j satisfies the Evans-Hudson equation djt = jt ° 02 dA£J(<), in which [0p] is the matrix of components of 9 with respect to the basis of k used for defining the matrix of quantum stochastic integrators [A|]. Furthermore, if j is *-homomorphic then the quantum Ito formula implies that 6 satisfies 0(a*a) = 0(a)*i(a) + t(a)*0(a) + 0(a)*A0(a),

(2)

where t(a) = a®/j, A = If, <8> P and P is the orthogonal projection in B(k) with range k C k. Conversely, (2) entails complete boundedness of 0, which implies that 0 generates a regular Markovian cocycle j on ,4;10 in turn (2)

181

182

also implies that j is completely positive and contractive (Proposition 2 below). The question we address here is when is j multiplicative too, and so *-homomorphic? When k has finite dimension d, Evans3 proved multiplicativity by showing that the difference Jt(a, b, £, £) := Ot(a)£,jt(&K) - (£,jt(a*b)£) satisfies Jt(«,&,£,0 = 5"" L / a<(*)J,(&(a),lM&),£,C)d«, *— Jo

(3)

where N = (1 + d)2(2 + d), each fa and ^ is the identity map or one of the bounded operators Op, and each a, is locally square integrable. This may be iterated and then estimated to be shown to be vanishingly small. The challenge in the case of infinite dimensional k was first taken up by Mohari and Sinha.14 Their essential idea was to make the following judiciously chosen relative boundedness assumption on 9: For all (3 there is (K,L>) s.t. Va,u \\0(a)E(p)U\\ < \\(a® IK)Du\\. (The operator E^ is defined below.) With this they were able to tame the plague of infinitely varying indices arising from iteration of (3), and again show that the result becomes vanishingly small. The Mohari-Sinha assumption is perfectly tailored to the case where A is a von Neumann algebra. The reason for this is that (due to the necessary complete boundedness of 6) their assumption is equivalent to the ultraweak continuity of d. On the other hand, in the C""-context their assumption now appears ad hoc. Here we exploit the compete boundedness of both generator and cocycle to give a simple and short proof of multiplicativity under two alternative hypotheses. The first of these, which is new, is natural in the C*-context. We apply it to give a direct proof of a dilation theorem due to Goswami, Pal and Sinha.4 The second is natural in the von Neumann algebra case. The first hypothesis is satisfied when the noise dimension space is finite dimensional; both hypotheses are satisfied when the algebra is finite dimensional. A deeper study of the problem has been undertaken elsewhere.12 There necessary and sufficient conditions on 6 are found by means of a product formula for iterated quantum stochastic integrals (cf. the work of Hudson and co-authors5). Alternative additional hypotheses on 6 are then given for (2) to entail these conditions. The question of whether (2) alone implies multiplicativity without additional hypotheses remains open.

183

General notation The algebraic tensor product is denoted 0, the Hilbert space tensor product ®, and the spatial and ultraweak tensor products of concrete operator spaces (closure in the norm and ultraweak topologies respectively) are denoted by <8>sp and ®uw- Tensor symbols between Hilbert space vectors will be dropped. If H and h are Hilbert spaces and x € h, the map u i—> ux from H to H ® h will be denoted Ex, with the particular choice of H determined by the context; the adjoint of Ex is denoted Ex, and when an orthonormal basis (ej), e / for h is understood, E^ denotes Ex for x = BJ, and similarly for E^ . When H = C the notations \x) and (x are also used, and the (operator) spaces consisting of these operators are denoted |h) and (h| respectively. Preliminaries Throughout this note ,4 is a unital C"*-algebra acting on the Hilbert space f), k is a separable Hilbert space, and J- denotes the symmetric Fock space over L 2 (R + ; k). The Hilbert space k is defined to be C © k, the noise space augmented by a copy of C. Operator spaces: matrix, row and column spaces Let V be a concrete operator space2 on a Hilbert space H, that is a closed linear subspace of 5(H). For any other Hilbert space h we define the hmatrix space over V to be M(h;V) b := {A £ J3(H ® h) : EdAEe € V V d . e e h } . Since the map e i—> Ee is (completely) isometric, to check that an operator A in 5(H ® h) belongs to M(h; V) b it suffices to check that EdAEe £ V for all d and e from some total subset 5 of h, such as an orthonormal basis (ej)i 6 /. Such a choice of basis leads to the identifications M(h; V) b S M/(V)

A <-^ [Aj], A] := E^AE(j},

which explains our terminology. Rectangular matrix spaces M(h 2 , hi; V) b are defined similarly. In particular the row and column space of V over h are defined by

R(h; V)b := {A € 5(H ® h; H) : AEe S We e h}, C(h; V) b := {A e B(H; H ® h) : EdA £ VVrf e h}.

184

Matrix spaces lie between more familiar tensor products:

V ®sp 5(hi; h 2 ) c M(h 2 , hi; V) b C V ®Uw 5(h x ; h 2 ), where the first inclusion is an equality if either V, or both h j and h 2 , are finite dimensional, and the second is an equality if and only if V is ultraweakly closed. In particular, C(h;,4) b lies beween the Hilbert C*module *4®Sp |h) and the Hilbert TV-module A" <8>uw |h). Let : V —» W be a completely bounded (CB) map into another operator space. Then, for any Hilbert space h, induces10 a unique CB map ^ : M(h; V) b -* M(h; W) b satisfying Ed^h\A)Ee = (EdAEK)

\/d,e € h,A 6 M(h;V) b ,

and moreover, ||<^||cb = ||$||cb- The same goes for rectangular matrix spaces such as row and column spaces. The map Js appearing in (1) is an example of such a <j>^, where V = A, ja is a CB map A —> M(^rs; ,4)b, and h = J-3 , where fs and J-s are the symmetric Fock spaces over £ 2 ([0, s[; k) and L 2 ([s,oo[;k) respectively. This construction will be used repeatedly below in a tweaked version when the CB map in question has a matrix space as its target: if : V —> M(h';V) b isCBthen ^h:M(h;V)b-+M(h';M(h;V)b)b is the map obtained by composing (j>^ with the natural completely isometric isomorphism M(h;M(h';V) b ) b -M(h';M(h;V) b ) b that is obtained by restriction of the flip B(H ® h' h) -> B(H h ® h'). In particular we will write (j)"n when h = k® n , and simply *when n — 1. Thus jg is a map

and not the map j~s discussed previously. The doffing of the hat, ladies and gentlemen, is not some sleight of hand designed to pull the wool over the reader's eyes. The following result shows that a useful property for columns of a CB map <j> : V —» M(h'; V) b is inherited by the induced maps (j>h. Lemma. Let h and h' be Hilbert spaces, let : V —> M(h';V) b be a completely bounded map satisfying 4>(a)Exi € V ®Sp |h'} for all x' & h' and

185

a e V, and let A £ M(h; V) b be such that AEX £ V ®Sp |h) for all x e h. Then 4>h(A)Et 6 V <8>Sp |h <8> h'} for all £ e h ® h'. Proof. Since the map £ i—> .Eg is isometric it suffices to prove that 0 (h) (^)£ I - I £ V ®sp |h' ® h) Vz 6 h, x' e h'. If (A< h > is the CB map C(h; V) b -> C(h;M(k; V) b ) b induced by <£ then

Moreover, for any a e V and y 6 h, W(a®\y))Ex. = (a)EX'®\y). Therefore, by the assumptions on (j>

an

d A,

EX, e V ®Sp |h) ®sp |h') = V ®sp |h ® h'). D Markovian cocycles Following our recent philosophy,11 we concern ourselves here with completely bounded processes on an operator space V, that is time-indexed families j = (jt)t>o of CB maps from V into B(H f), where H is the space on which V acts. Such processes are required to be measurable: t >-> (£, Jt(a)C) is measurable Va £ V,£, C € H ® F, and adapted:

Although our processes will be everywhere defined we will still make use of the following exponential domain: £ := Lin{e(f) : f e §}, where S := Lin{cl[0)t] : c e k, t > 0}. Following the notation of (2) we write

4 h (o) := a ® /h, AH := IH ® P,

(a & V),

(4)

dropping the subscript on <, and A when h = k (resp. H = f)). A CB process j is a Markovian cocycle on V if it satisfies (1), and furthermore is a regular cocycle if its Markovian semigroup (Ee^jt(-)E£^)t>o is norm continuous. If the cocycle acts on the C*-algebra A and is both regular and composed of completely positive (CP) contractions then there

186

is a CB map 6 : A —> M(k; A)-^ such that j strongly satisfies the quantum stochastic differential equation (5)

Here the maps 9% are the components of 9 with respect to the orthonormal basis (e a ) a >o of k (in which CQ = 1 € C) with respect to which the matrix of quantum stochastic integrators is defined. Conversely, any CB map 9 : V —> M(k; V)^ gives rise to a strong solution of (5) that is a Markovian cocycle. (Since the cocycle need not consist of bounded maps, let alone CB maps, the cocycle definition must be extended appropriately.)9 Suppose that a CB map 9 : V —> M(k;V)],j generates a CB cocycle j = (it '• V —> M(J~t', V)b)t>o- In this context the first fundamental formula of quantum stochastic calculus13'15 gives the following basis independent identity: <«£(/), (jt(a) - ^(a]\v£(g)) (g)) = I (uf(s)e(f),j;(0(a))vg(s)e(g)) (uf( ds (6) Jo for all u,v e H, /, g € S, a € V and t > 0. Here f ( s ) and
=

/o (7)

where £(s) = uf(s)e(f), £(s) = v'g(s)s(g), and t and AH are as in (4). The next result is summarised by the commutative diagram

01

in which horizontal arrows denote generation of Markovian cocycles. Proposition 1. Let 6 : V —» M(k; V)^ be a completely bounded map that generates a completely bounded cocycle j. Then for any Hilbert space h the (completely bounded) map 9h generates the cocycle jh on M(h; V) b .

187

Proof. The following identity is easily verified ds,

for £, £ £ H0 h, and consequently for all vectors in H ® h. In view of (6) and the relation jj® k = (j£)", the result follows from the uniqueness of weakly regular weak solutions of quantum stochastic differential equations.8 D The Homomorphic Property As explained in the introduction, the identity (2) is necessarily satisfied by generators of regular *-homomorphic cocycles. Previous work on regular CP contraction cocycles,7'8 summarised in the following result, implies that such a map 9 is both completely bounded and the generator of a CP contraction cocycle. Proposition 2. Let 9 : A —> M(k;^)] 3 be a linear map satisfying (2). Then 9 is completely bounded and generates a completely positive contraction cocycle. Proof. The identity (2) implies that 9(a) has the block matrix form

where ?r is a "-homomorphism A —> M(k; < 4)] 3 , 5 a 7r-derivation A —» C(k;«4)jj, and 5^(a) := 6(a*)*. Furthermore, several applications of Corollary 2.3 of Christensen and Evans' paper1 leads to the decompositions S(a) - 7r(a)L - aL,

r(a) = L*Tt(a)L - \{a, L*L} + t[a, h]

for some L e C(k;A")}j and h = h* e ^4", revealing that 9 is completely bounded. Let j denote the Markovian cocycle generated by 0, and let <j> be the map a H-» 9(a) + a P. Then, for any (m, /i),..., (u n , /„) E I) x S and A = [Oj] G Mn(A), the semigroup representation of Markovian cocycles7'8'9 gives

(9)

188

where {0 = to < • •• fc := Y^<j)^(-)Yk, in which Yk = diag[£/i(t)c), . . .,E?n(tk)] e M n (B(t); f, 8 k)). By (2),

/ij 8P-1 8/c-)tf (n) (4) (^ ® P X ) + «(A 8 P-1) V (n) (A)

(10)

for all n > 1 and A 6 Mn(./4), with K denoting the flip automorphism 5([)<8>Cn<8>k) —> B(f)<8>kk now follows from the identity K,(A®PL)Yk = diagfEg, . . . , E$]A, and thus each semigroup Pk is positive. By a simple re-indexing trick, (9) now implies that the cocycle j is completely positive. To prove contractivity it suffices to show that the positive operator j t ( l ) is a contraction. Now (2) implies that 9(1) = -0(l)*(Il><S>P)9(l) < 0. But, for ip = £?=1 Uje(/0 £ l ) OS,

where tp € (I) 0 £)n is the vector with ith component tt»£(/i) and Z(s) = [E^ (a) • • • Ej~ (s}] 6 B (f) n ; () ® k) . The result follows. D Thus any map 6 : A —> M(k; A)^ satisfying (2) is completely bounded and generates a CP contraction cocycle; our first proposition therefore applies. Before reaching our goal we require a third and final device, which is that (under reasonable conditions) the polarised form of (2), namely 6(ab) = 0(a)i(6) + i(a)0(b) + 0(a)A0(b) Va, b & A,

(11)

remains valid when we pass to the h-matrix space over A. For finite dimensional h the result is easy, and has already made a veiled appearance in (10). Proposition 3. Let 0 : A —> M(k; A)^ be a completely bounded map satisfying (11), and let h be a Hilbert space. (a) Suppose that B € M(h; A)-^ satisfies BEX e -4<8>sp |h) for all x e h. Then, for all A e M(h;.4) b , AB e M(h;^) b and 9h(AB) = 6h(A)i(B) + i(A)9h(B) + 0 h (A)A^ h 0 h (S).

(12)

189

(b) Suppose that A is a von Neumann algebra and 9 is normal. Then satisfies (12) on all o/M(h; «4)b = .4<8>uw 5(h). Remark: Notice that, in our versatile notation, i coincides with t h . Proof, (a) Let x, y e h and x, $ € k, then ExABEy = (A*Ex}*BEy e R(h;.4) b (.A®sp |h» = A and so AB £ ^^(h;^)]-,. Let {/7}7ej be an orthonormal basis for h. Thus £7 £(7)£(7) and £7 E^&E™ converge strongly to 7^h and A 7h respectively. But also, for each y € h, since jB£H £ .4 ®sp |h) it follows that ^7 E^E^BEy converges in norm to BEy. Thus, by the norm continuity of 9, and (11), Ex9h(AB}Ey = B(EXABEV)

It follows that (12) holds. (b) This case is simpler. It is easily verified that #h satisfies (12) on A(D B(h), and both sides of the identity are separately ultraweakly continuous in A and in B. The result follows. . D Our trio of propositions can now be exploited in the main act. Theorem. Let 6 : A —> M(k; A)-^ be a linear map satisfying (2). Then 6 is completely bounded and generates a completely positive contraction cocycle j which, under either of the following conditions, is * -homomorphic: (a) 9(a)Ex € A ®sp |f<) for all a e A and x € k. (b) A is a von Neumann algebra and 9 is ultraweakly continuous. Proof. In the spirit of the original proofs of multiplicativity3'14 we shall obtain vanishing estimates, through iteration, for the difference («e(/), [jt(ab) - jt(a)jt(b)}v£(g)}.

(13)

Note that here, by the second proposition, we already know that 9 is completely bounded and that jf is a CP contraction cocycle, so that the product jt(a)jt(b) above is well defined. Also by the first proposition, for each n > 1, the map 9"n generates the CB cocycle j"n on M(k®"; A)^. Setting

190

gives the identity (j - rr=j ;(«+D

vn>0.

(14)

The following CB maps ¥>„,,-, Vnj : M(k®";.4)b -» M(k®("+1);.4)b (n > 0, j G {1,2, 3}) appear in the iteration:

(a) For each n define Sn = {A € M(k®"; A)b :AE&£A® |k® n ) VS e k®"}. Part (a) of Proposition 3 implies that if A 6 M(k8>"; .4)b and B e 5n then ABeM(k® n M) b and

So applying the consequences (6) and (7) of the first and second fundamental formulae, and identity (14), to the difference

yields

where A» = V«,t(-4) and 5, = V> n ,»(-B). But 5 and i satisfy the hypothesis of the lemma in the previous section, and so for any b S A, n > 0 and i0, . . . , in € {1,2, 3}, V ' n , i n o - - - o V o , i o ( 6 ) ^ e ^ ® s p | k ® ( n + 1 ) ) V<5€k® ( " + 1 ) .

(15)

The above identity can therefore be iterated from n = 0 in order to estimate (13). After n steps an upper bound for the modulus is given by 3" integrals over the simplex {s G M™ ; t > si > • • • > sn > 0} each of whose integrands is of the form |(U/®"(s)£(/),

(j;nn(AB)-js"nn(A)j;nn(B)}vg®n(S)£(g))\,

where A is of the form o, »0 (a) an(i B is of similar form, with the maps y>n,i acting on b. Now j"n is a contraction process and 9 is completely bounded, so the modulus of (13) is bounded above by 3n(n\)~1CiC2 where andC 2 =

191

Hence j is multiplicative, and thus *-homomorphic. (b) The proof in this case is identical except that part (b) of Proposition 3 should be employed rather than part (a), and the subspace Sn no longer plays a role, since (15) is replaced by

A ®uw 5(k® (n+1) ).

n

A Dilation Theorem Revisited Part (a) of the theorem covers the cases of finite dimensional k (Evans' original result) or A. It also facilitates the following short proof of a dilation result of Goswami, Pal and Sinha4 for quantum dynamical semigroups. Moreover this new proof avoids having to pass to the universal enveloping algebra and subsequently show that the (normal *-homomorphic) process constructed there leaves the C"*-algebra "invariant." Theorem. Let P = (Pt)t>o be a norm continuous completely positive contraction semigroup on the C* -algebra A, and suppose that A is separable. Then P has a * -homomorphic stochastic dilation: there is a regular *homomorphic cocycle j with separable noise dimension space k such that

Proof. Let r be the generator of P. The first part of the proof is by now standard:1'7 since P is completely positive and contractive, r(a*) = r(a)*, r(l) < 0, and (a, 6) K-* r(a*6) - a*Y(6) - r(a)*6 + a*r(l)6 defines a nonnegative definite kernel k : A x A —» .4 C B(f)). Let (K, 7) be its minimal Gelfand pair, thus 7 is a map A -> S(f); K) satisfying 7(o)'7(i) = k(a, b) and K = Vm-y(A)\). The sesquilinearity of k and minimality of (K,7) imply that 7 is a linear map, and thus a bounded operator, and that there is a unique unital representation a of A on K satisfying (7(0)7(6) = 7(06) — 7(0)6. Thus 7 is a cr-derivation. Next, following Goswami, Pal and Sinha, let F = Lin{7(a)i> : a, b 6 A} C £((); K) and note that F is a Hilbert ^-module, with A- valued inner product defined by (, ip) = <j>*ip. Separability of A implies that F is countably generated, and so Kasparov's Absorption Theorem6 implies that there is a separable Hilbert space h and an adjointable isometry a : F —> A ®Sp |h) C -B(h; f) h). The set of vectors {<jyu : $ € F,u € ()} is total in

192

K, and moreover {(f>u,<j>'u') = (a((j>)u,a(')u'). V : K —> t) <8> h satisfying

Hence there is an isometry

In particular, Va(a)V*Ex = Va(a)a*(Ex) = a(a(a)a*(Ex)) 6 A ®sp |h>.

(16)

Now let k = h ©C and define TT : A -» 5(1) ® k) and S : A -> 5(1); h ® k) by (a)V*O

V

o

where d = (— r(l)) 1 / 2 . Defining 0 in terms of r, 5 and TT through (8) then gives a map A —> B(l) <8> k) that satisfies (2). Moreover, it follows from (16) that 0 also satisfies 9(a)Ex € A ® |k). Therefore, by the theorem, the Markovian cocycle generated by 0 is *-homomorphic. It is a stochastic dilation of T3 since the semigroup (£ l£ ( 0 ^jt(-)E £ ( 0 ))t>o has generator T. D Remark. A less pedestrian use of Hilbert modules would shorten the proof even further. Acknowledgement We are grateful to Debashish Goswami for useful conversations and a careful exposition of the dilation theorem4 during his visit to Nottingham in July 2001. SJW acknowledges the support of a Lloyd's Tercentenary Foundation Fellowship. References 1. E Christensen and D E Evans, Cohomology of operator algebras and quantum dynamical semigroups, J London Math Soc 20 (1979), 358368. 2. E G Effros and Z-J Ruan, "Operator Spaces," Oxford University Press, 2000. 3. M P Evans, Existence of quantum diffusions, Probab Theory Related Fields 81 (1989), 473-483. 4. D Goswami, A K Pal and K B Sinha, Stochastic dilation of a quantum dynamical semigroup on a separable unital C*-algebra, Infin Dimens Anal Quantum Probab Relat Top 3 (2000), 177-184. 5. R L Hudson, Unitarity and multiplicativity via higher Ito product formula, Tatra Mt Math Publ 10 (1997), 95-108.

193

6. E C Lance, "Hilbert C*-modules," London Mathematical Society Lecture Note Series 210, Cambridge University Press, Cambridge, 1995. 7. J M Lindsay and K R Parthasarathy, On the generators of quantum stochastic flows, J Fund Anal 158 (1998), 521-549. 8. J M Lindsay and S J Wills, Existence, positivity, and contractivity for quantum stochastic flows with infinite dimensional noise, Probab Theory Related Fields 116 (2000), 505-543. 9. J M Lindsay and S J Wills, Markovian cocycles on operator algebras, adapted to a Fock filtration, J Funct Anal 178 (2000), 269-305. 10. J M Lindsay and S J Wills, Existence of Feller cocycles on a C""-algebra, Bull London Math Soc 33 (2001), 613-621. 11. J M Lindsay and S J Wills, Completely bounded Markovian cocycles on operators spaces, Preprint (2001). 12. J M Lindsay and S J Wills, Homomorphic Feller cocycles on a C*algebra, Preprint (2001). 13. P-A Meyer, "Quantum Probability for Probabilists," 2nd Edition, Lecture Notes in Mathematics 1538, Springer-Verlag, Heidelberg, 1993. 14. A Mohari and K B Sinha, Quantum stochastic flows with infinite degrees of freedom and countable state Markov processes, Sankhya Ser A 52 (1990), 43-57. 15. K R Parthasarathy, "An Introduction to Quantum Stochastic Calculus," Birkhiiuser, Basel, 1992.

A RELATION BETWEEN THE GROSS LAPLACIAN AND TIME CHANGES ON BROWNIAN MOTION

NICOLAS PRIVAULT Universite de La Rochelle, Avenue Michel Crepeau, 1704S La Rochelle Cedex 1, France E-mail: [email protected] We show that the transformations of random functional by time changes on Brownian motion can be expressed as the adjoints of generalized Fourier-Mehler transforms. The derivatives of one-parameter families of such transformations of Brownian functionals are computed using a weighted Gross Laplacian and a second quantized operator.

1. Introduction The Fourier-Mehler transform6 is a group (J-e)e^K of transformations of random variables that provides a natural analog on Gaussian space of the Fourier transform. The adjoint of the Fourier-Mehler transform also forms a group whose infinitesimal generator is the sum of the Gross Laplacian and the number operator. It has been extended as a two-parameter family of transformations2 and as families of transformations1,3 indexed by continuous mappings on <S(R). Moreover it has been noticed2, Cor. 4.4, that complex dilations of Gaussian measures can be expressed via the adjoint of the Fourier-Mehler transform. In this paper we show that transformations of Brownian functionals by time changes on Brownian motion can be expressed as the adjoints of generalized Fourier-Mehler transforms. Although such transformations are not quasi-invariant they can be defined on a dense linear space of smooth Brownian functionals. We compute the infinitesimal generators of one-parameter families of such transformations using a generalized Gross Laplacian and second quantized operators. These relations are viewed as the infinitesimal statement of an Ito formula without adaptedness requirements. In Sect. 2 we review the tools of white noise analysis that will be used in this paper. In Sect. 3 we define a family of transformations of random variables by time changes of Brownian motion. The generalized Gross

195

196

Laplacian is introduced in Sect. 4. An expression of transformations of random functional by time changes on Brownian motion is given in Sect. 5, using the adjoint of the generalized Fourier-Mehler transform. In Sect. 6 we determine the infinitesimal generators of families of such transformations using a weighted Gross Laplacian and second quantized operators. 2. Notation and preliminaries Let <S(R), <S'(R) denote respectively the Schwartz spaces of test functions and distributions with pairing < •, • >, and let (-,-), • denote the scalar product and norm in L 2 (R+). The white noise space (iS'(R), n) is equipped with the standard Gaussian measure /j, on <S'(R) defined as r /

1 e x p ( t < x , £ > ) d / i ( a O = e x p (--

JS'

Let L 2 (R+) be the space of symmetric square-integrable functions on R" . We denote by /„ g^ the tensor product of /„ € L2(W±), gm e. L 2 (R^), and by u\ o- • -oun & L 2 (K") the symmetrization of M I ® - • - ® u n 6 L 2 (R") in n variables. Let C£°(R+) denote the space of C°° functions on R with compact support in R+, let 3- denote the cr-algebra on <S'(R) generated by x H-* (o;,0, f e CC°°(R+), and let (L 2 ) = L 2 (5'(R),^,//). Each F 6 (L 2 ) has a decomposition

F=

/„(/„),

/ n GL 2 (R!^),

(1)

n=0

where In(fn) is the multiple stochastic integral of the square-integrable symmetric function /„ G L 2 (R" ) of n variables with respect to the standard Brownian motion (B(t))te^+ denned as B(t) =< x,l[0,t] >, x € <S'(R),

Definition 2.1. Let P denote the space of square-integrable random variables of the form

«!,...,«„ eC c ° For y £ 5'(R), the gradient Dy is denned on P as

197

For t 6 R, the white noise gradient dt is denned as dt = D$t, where St is the Dirac distribution at t, i.e.

n=l

if .F e P is written as in (1). Let (5) and (S)* denote the white noise spaces of test functions and distributions. The operator dt extends as a continuous operator from (S)* into (S)* and its adjoint is denoted by <9(*, t G R + . The operators dt and 8% are linked by the relation

where B(t] is the white noise. The exponential vector 4>£ is defined as

and we let S denote the vector space generated by {(j>£ : £ 6 C which is an algebra contained in P and dense in (L 2 ). Finally, given a mapping A : L 2 (M+) —» L 2 (R+) we let F(j4) denote the second quantization of A, defined on S as

3. Time changes on Brownian functionals In this section we define a family of transformations of random functionals by time changes on Brownian motion. Let C0°°(R+) = {h £ C°°(R+) : h(0) = 0 and lim h(t) - +00}. t—too

Definition 3.1. Given v € C£°(R+), we define Rv : C^°(R+) —> C~(R+) and A(^) : P —> ^ as

and

Since Rv is not continuous we need to show that this definition is independent of the particular representation F = /(/i(wi), . . . , /i(u n )) chosen for Ftp.

198

Prop 3.1. Let F, G e P be written as F = f(Ii(u1),...,I1(un)),

ui,... ,«neC~(R+),

and

If F = G a.s. then A^F = A.(RV)G, a.s. Proo/. Let ei, . . . , e/t € C£°(R+) be orthonormal vectors in I/ 2 (R + ) that generate MI, . . . , un, vi,..., vm, with m = X^l" °^ej and u* = Z)j-=" Pi ejThen F and G are also represented as F = /(/i(ei), . . . , /i(efc)), and G = fi(/i(ei),...,/i(efc)), with =fc

, • • • , Vk) = f

j=fe

\ a

y

J2 "1%' ' ' ' ' ^2 n i

' J/i. • • • , 3/fe

and /3iyj,...,^2/j

, yi,...,j/fc 6R.

Since F = G and /i(ei), . . . ,/i(efe) are independent, we have f = g. Moreover by linearity of /i and .Rj/ we get

and

hence A(^)F = A( J R I/ )G.

D

Next we show that if j/ G Co°(R+) is strictly increasing, then the action of A(RV) is to evaluate a given smooth functional on time-changed trajectories

Prop 3.2. We have for F = /(/i(ui), . . . ,/i(u n )) 6 ^:

a

00

,.00

^^^^(i/- 1 ^)), . . . , / Jo

\

un(t)dB(v-l(t))

/

, a.

199

Proof. Since u» and RvUi are C£° functions, the stochastic integrals /0°° Ui(t)dB(i) and /0°° Ui(i>(t))dB(t) can be denned for every path of />OO

OO

/

Ui(v(t))dB(t)

=/

Jo oo

/-00

j4(*X.(z/(*))£(0* = Ui^dB^'1^)),

Jo

u't

a.s., i = l,...,n.

It remains to use the multiplicativity of A (./?„) which follows from Prop. 3.1:

4. Generalized Gross Laplacian Given a mapping /^ : <S(K) —> <5(R), let r(K) denote the trace operator associated to K, and denned as

Let AG(K) denote the generalized Gross Laplacian associated to K} cf. Def. 3.1. of Chung and Ji1, defined here on P as

= I2 Jr

T(K)(s,t)dsdtdsdt,

i.e.

In this section we introduce a particular generalization of the Gross Laplacian. Let h € C°°(R+), and let Kh denote the operator Kh : C£°(R+) —> C£°(1R+) defined as Kh£(t) = h(t)?(t),

t 6 R + , £ e CC°°(M+).

Definition 4.1. Let t € R+. We define 5^ on P as S/ = --D^i, where Si is the first distributional derivative of St, i.e. i

d

The generalized Gross Laplacian Ac(^h) associated to JCfc can be expressed as oo h(s)dsdlaFds. (2) /

200

The following proposition expresses AG(^'/I) as a weighted Laplacian. Prop 4.1. Let h 6 C£°(R+). We have 1 I"00 AG(Kh)F = -h'(s)dadaFds, ^ Jo

Proof.

F£P.

(3)

For F = /(/i(«i), . . . , /i(u n )) 6 P we have

s, t > 0. Hence by integration by parts on R+, using the condition h(0) = 0, (I\

F = y^ (h Uju')

I v - ^ . . ^ d*f D

5. Time changes and the Fourier-Mehler transform We now present a relation between the operator A.(RV) and the adjoint of a generalized Fourier-Mehler transform. This relation can be viewed as the integrated form of the relation proved in the next section. In the case of complex dilations of Gaussian measures this type of result has been obtained in Cor 4.4-(v) of Chung and Ji2. For A and B two continuous linear mappings on <S(R), the transform Q(A,B) has been defined in Lemma 4.1 of Chung and Ji1 as

and shown to be equal to the adjoint of a generalized Fourier-Mehler transform. We note that this definition is still possible on the space H without continuity assumptions on A, B : C£°(R+) —> C£°(R+), since the expression of F £ E as <*i^, 1=1

is unique whenever & ^ £j, i ^ j.

(4)

201

Prop 5.1. Assume that v 6 Cg°(]R+) is bijective on R+ and written as for some h £ C°°(R+). Then A(fi^) = Q (-Kh, Rv). Proof.

We will prove the following relation on H: A(^) = T(RV) exp (-Ao(^)) -

(5)

We have

= exp

-

exp

= exp /

/-o

= exp - / V -/o

n Similarly, if /i € C°°(R+) satisfies i/(t) = t + /i(t), t € R+, we can prove that A(fl tf ) = exp (A G ( as follows:

= exP = exp

o = exp (Ac

i/wwwwnw) - t}dt

202

6. Derivatives of transformations induced by time changes It has been shown4 that the generator of the adjoint of the Fourier-Mehler transform is given by the sum ds d*ds + l- [ ds dsds * Jo

i I Jo

(6)

of a second quantized operator (the number operator) and the Gross Laplacian. The formulas 7-10 obtained below are an extension of these results one-parameter families (A(.Re/l))e€R of transformations of Brownian functionals induced by time changes on (B(t)) t6 K + . The operator /0°° ds d*d] in the next proposition is in fact the differential second quantization of the operator of differentiation of Fock kernels, which differs from the number operator /0°° ds d*ds. Prop 6.1. Let h e CC°°(K+). For all e > 0, define ve e Cg°(R+) as

Then e »-» A(J?t/JF is differentiable in (L2) for all F e P and we have the equalities: £=0

= dT(Kh) + AG(Kh) poo

oo

ds h(s}d*sdls +

/

(7) ds h(s)dsd1s

Jo

= Jor ds h(s)d;dl -\4 Jr ds h'(s)dsds, )

(8)

(9)

0

ds h(s)B(s)d1s.

(10)

Proof. On the space E! of exponential vectors these formulas follow directly by differentiation of (5) under the hypothesis of Prop. 5.1. We need to proceed differently in order to do the proof on P. We will show that (7

f°° 1 r°° JF| £=0 = / h(s)dtdl,Fds - - I ti(s)dsdsFds. Jo 2 JQ

(11)

The expressions (7) and (9) follow then from Prop. 4.1, and (10) follows from the relation B(t) = d* + dt. a) We start by proving (11) for F = I^u), u e C£°(R+). Given that V /D \ lim —(Rv eu — u)

f. / = + nu ,

203

we have A(^JF|£=0 ( = 2)

/!(&,,. «)| e =o - /i(/m') -

°° h(s}d*su'(s)ds

since &.G(Kh)Ii(u) = 0. b) Next we show the chain rule

. . . , Fn)

(12)

Fi,...,FneP, f e C6°°(Rn), using the relation We have

h ( s ) ( d l s F i ) d s f ( F 1 , . . . , Fn)ds

1

(13)

On the other hand, using the fact that 9S is a derivation operator on T7 it can be shown exactly as for the classical Gross Laplacian5, Th. 6.18, that ...,Fn}

t=i

(14)

204

i i±i? ppf

r°°

Fl -oz E u;nH """F")/ (W(WAs)rfs. (15) j w_i yiuyj Jo

Combining (13) and (14) we obtain (12) on P. Since J0°° h(s)d*dlds + Ac(Kh) and dA(J?t,e)/d£|e=o satisfy the same chain rule of derivation and coincide on first chaos random variables, they coincide on P. D If h is defined by h(t) = -t, t & K+, then AG(Kh) = AG is the classical Gross Laplacian. On the other hand, the operator &.c(Kh) can be interpreted as an infinite-dimensional realization of the generator of Brownian motion: indeed for all fixed T > 0 and h € C£°(R+) such that h(T) = -1 we have the relation

=

f"(B(T)),

f e C62(R).

The computation of the derivative of one-parameter families of transformations associated to time changes:

r°° 1 f°° = \ ds h(s)d*sdls-- / ds h'(s)dsds, Jo * Jo can be viewed as an elementary non-adapted Ito formula in which the finite variation term and the stochastic integral term correspond respectively to the Gross Laplacian and to the second quantization of the derivation of Fock kernels. d

— "•£

References 1. D.M. Chung and U.C. Ji. Transforms on white noise functionals with their applications to Cauchy problems. Nagoya Math. J. 147, 1-23 (1997) 2. D. M. Chung and U.C. Ji. Transformation groups on white noise functionals and their applications. Appl. Math. Optim. 37, 205-223 (1998) 3. D.M. Chung, U.C. Ji, and N. Obata. Transformations on white noise functions associated with second order differential operators of diagonal type. Nagoya Math. J. 149, 173-192 (1998) 4. T. Hida, H. H. Kuo, and N. Obata. Transformations for White Noise Functionals. J. Fund. Anal. Ill, 259-277 (1993) 5. T. Hida, H. H. Kuo, J. Potthoff, and L. Streit. White Noise, An Infinite Dimensional Calculus. Kluwer Academic Publishers, Dordrecht, 1993. 6. H.H. Kuo. Fourier-Mehler transforms of generalized Brownian functionals. Proc. Japan Acad. Ser. A Math. Sci. 59, 312-314 (1983)

A NOTE ON BOSE 3-INDEPENDENT RANDOM VARIABLES FULFILLING Q-COMMUTATION RELATIONS

MICHAEL SKEIDE Lehrstuhl fur Wahrscheinlichkeitstheorie und Statistik, Brandenburgische Technische Universitat Cottbus, Postfach 10 13 44, D-03013 Cottbus, Germany, E-mail: skeideQmath. tu-cottbus. de, Homepage: http://www.math.tu-cottbus.de/INSTITUT/lswas/-skeide.html In Reference17 Voiculescu generalizes his notion of free independent random variables16 to the notion 3-free 3-random variables (free independence with amalgamation over 3; see also Speicher15). Surprisingly, an amalgamated version of Bose independence resists to be meaningful in full generality (roughly speaking, because there is in general no tensor product of 3-random variables). In these short notes (a slightly revised version of the preprint10) we intend to do not much more than to propose a definition of Bose 3-independence at least for so-called centered 3-random variables and to present some examples including creators and annihlilators on the symmetric Fock module11 and some random varibles fulfilling q-comutation relations. The crucial notion of centered 3-random variables relies on the notion of centered Hilbert modules11 and it can be shown that every !B(Gr)-random variable (G some Hilbert space) is of that type; see References13'2. Meanwhile, we know that every central limit distribution of Bose 3-independent 3-random variables may be represented by creators and annihilators on some symmetric Fock module; see Reference12.

1. Basics Throughout these notes 3 denotes a C"*-algebra. (Using the algebraic methods from l also more general *-algebras are possible. See Appendix C of Reference14 for a systematic introduction to such P*-algebras.) Algebras are unital and modules are modules over algebras. All tensor products and direct sums are algebraic. The word 'centered' is reserved for a certain type of two-sided module or elements in the 'center' of such a module (see below). 205

206

A random variable whose first moment vanishes will be called 'mean-zero random variable'. The tensor product over 3 of two 3~3-rnodules E and F is the 3-3-module denned by E 0 F = E F/(xz ® y — x <S> zy). By £ and S we denote spaces of linear and bounded linear mappings, respectively. A superscript ° means (mutually) adjointable mappings. Notice that adjointable mappings on inner product modules are right linear automatically. 1.1 Definition. A ^-algebra is a 3-3-module A with a 3-3~h'near multiplication M: A 0 A —» .4 and a 3~3-linear unit mapping m: 3 —> .4, such that the associativity condition

M o (M 0 id) = M o (id 0 M) and the unit property M o (m 0 id) = id = M o (id 0 m)

are fulfilled.

We use the notation M(a © b) = ab and m(l) = 1 (i.e.

13 = U = !)• A * -3 -algebra is a 3~algebra with a 3~3~anti-linear involution, i.e. (zaz')* = z'*a*z* (a 6 A; z, z' & 3). A Jt-state is a positive normalized (i.e. y(l) = 1) 3~functional (i.e. a 3-3^1inear mapping y>: A —> 3)A 3 -quantum probability space is a pair (.A,
207

1.3 Definition. The ^-center of a 3-3-module E is the set {x 6 E:xz = zx (z € 3)}- A 3-3-module E is called centered, if it is generated by its 3-center. This means that for any x 6 E there exist n e N, Xk e Cs(E), zk 6 3 (A; = 1, . . . , n), such that

fe=l

= zkxk. fc=l

1.4 Proposition. 1. A 3~3 -linear mapping maps the 3 -center into the 3 -center. 2. Any element of a centered 3-3-module commutes with any element of the center of 3 . 3. Consequently, a 3-3-module over a commutative algebra 3 is centered, if and only if left and right action coincide. 4- For two centered 3~3 -modules E,F we have C^,(E] 0 C^(F] C ). Therefore, also E 0 F is a centered 3-3-module. 1.5 Theorem. Let E,F be two centered 3 ~3 -modules. There is a unique 3-3-fnodule isomorphism f . E © F —> F 0 E, called flip isomorphism, fulfilling

for all x e C5(E) and y € C3(F). The proposition is obvious. The proof of the theorem can be found in Reference11. Within the category of centered 3 -algebras (i.e. 3~algebras whose module structure is centered) we may define the tensor product AG>B of two 3-algebras A and B by defining the multiplication MAQB = (M^ 0 MB) o (id 0 y 0 id) and the unit IAQB = 1.4© IB- Obviously, y. A 0 B —> B 0 A defines a 3~algebra isomorphism. Notice also that the embeddings 6.4: a >-» a 0 1 and e&: b t—> 1 0 b define 3~algebra homomorphisms. 1.6 Definition. Let C be a subset of a 3-algebra C. By the 3-commutant C' we mean the 3~subalgebra of C generated by all elements of C which commute with all elements of C. If a 3-algebra C is its own 3-commutant C', we call it a 3-commutative 3-algebra. Notice that C" is usually much bigger than the C-commutant. If C is a centered 3-algebra and C3(C) is a commutative subalgebra of C, then C

208

is 3-commutative. For instance, in Reference4 a quantum dynamical semigroup on 3 is called essentially commutative, if it admits a dilation into a 3-commutative 3-algebra. The following universal property is checked easily: The tensor product of centered 3~algebras A and B is the unique centered 3-algebra A 0 B, such that for any pair j: A —> C and k:B—>C of 3-algebra homomorphisms into a 3-algebra C, fulfilling j(A) C k(B}' (and conversely), there exists a unique 3-algebra homomorphism j QM k = M o (j Q k): A 0 B —+ C, fulfilling (j ©M k) o eA = j and (j ©M k) o BB = k. (For C = 3 we have jQMk=jQ k.) Notice that all what we said remains true for centered *~3-algebras and *~3-algebra homomorphisms. In particular, we realize that j O k is a *-3~algebra homomorphism, if j and A; are. 1.7 Definition. A centered 3 -random variable B is a centered *-3~subalgebra of A. If we are interested only in centered 3-random variables, then we may restrict to the biggest centered 3-subalgebra A° of A. If Ac = A, we speak of a centered 3-quantum probability space. In this case, every a 6 A generates a centered *-3-subalgebra Bca of A. (Observe that Bca D Ba.) Again, we call an element a € A = Ac a centered 3-random variable, but, actually, we mean Bca. The notion of centered 3-random variable has not to be confused with mean-zero 3-random variables, i.e. a & A with on A to B. A family (Bi)i€l of centered 3-random variables in a 3-quantum probability space (A, Bn © • • • © VBin ,

where j$ (i £ T) denotes the canonical embedding Bi —> A. For 3 = C this is, clearly, the usual Bose independence of quantum random variables; see e.g. Schiirmann8. We see that the mixed moments of elements in Bose 3-independent 3-subalgebras of A equals the corresponding moments in the tensor product of these 3-algebras in the tensor product of their distributions. There are two other known notions of quantum independence. See Schiirmann9 for a unified description of all three cases.

209

Both of the other notions of independence are invariant (up to algebra isomorphism) under the exchange of their factors. However, in Reference11 we pointed out by an example that the 3~analogue of the tensor product of algebras is not symmetric under exchange of the factors. Actually, it does not even allow for an obvious definition of a multiplication. This is the reason why we have to restrict the category under consideration. We believe that the category of centered 3~algebras is the right one. (In this context, we neglect the rather obvious possibilities for twisted tensor products which always appear as graded modifications of Bose independence. In such a case the modules are no longer generated by their 3~center, but, by their even elements. In fact, the stochastic limit of the QED-Hamiltonian considered in Reference11 yields a module of this type.) 2. The symmetric Fock module 2.1 Definition. A pre-Hilbert ^-module over a Cr*-algebra 3 is a right 3~ module E with a sesquilinear inner product (•, *):E x E —> 3; such that (x,x) > 0 for x € E (positivity), that (x,yz) = (x,y)z for x, y 6 E; z € 3 (right linearity), and that (x,x) — 0 implies x = 0 (strict positivity). If («,«) is not necessarily strictly positive, we speak of a semi-inner product and of a semi-Hilbert ^-module. We remark that sesquilinearity and positivity imply ( x , y ) = (y,x)* (symmetry), and that right linearity and symmetry imply (xz,y) = z*(x,y) (left anti-linearity). A pre- (or semi-) Hilbert 3 -3 -module is a two-sided 3-3-module E which is also a pre- (or semi-) Hilbert 3-module, such that {x, zy) = (z*x, y} for x, y £ E; z G 3 (*-property). For two semi-Hilbert 3-3-rnodules E and F we turn the tensor product E&F into a semi-Hilbert 3-3-module, by setting (x 0 y, x' 0 y') = (y,(x,x')y'). For general reference on Hilbert modules we refer the reader to the book of Lance5. For an easy accessible introduction see Reference13. See also Reference13 for a more systematic investigation of centered semi-Hilbert 3-3-modules (i.e. a semi-Hilbert 3-3-module whose module structure is centered) also taking into account topological questions. In References13'2 we show that a (sufficiently closed) Hilbert 25(G)-B(G)-module (with a normal left multiplication) is isomorphic to a suitable closure of H ® 3 (see Section 3) and also the algebra of operators on such a module is centered. Therefore, literature (mainly dealing with S(G)) provides us with lots of centered modules.

210

The simple proof of the following proposition can be found in Reference13. 2.2 Proposition. If

E

is

a

semi-Hilbert

3 ~3 -module,

then

If E and F are centered semi-Hilbert ^-^-modules, then y.E®F —> F 0 E is an isometry. 2.3 Definition. Let E denote a semi-Hilbert 3-3-module. By the full Fock module F$(E) over E we mean the semi-Hilbert 3-3-module

(E°° = 3). On pT,(E) we define the creators l+(x) (x e E) by setting xn O...Qxi=xQxnO...G>x1,

i(x)l =x

and the annihilators i(x) (x £ E) by setting f ( x ) x n 0 . . . 0 x\ = (x, xn}xn-i 0 . . . O ii,

£(x)l = 0.

The creator and annihilator to the same x 6 E are adjoint elements of Ba(F-$(E}). Moreover, we have the relations

where the algebra element (x, y) acts as multiplication from the left. The above definition has been introduced by Pimsner7. However, notice that Pimsner only considers complete modules. The first use in quantum probability occured in Speicher15. The full Fock module F3 (E) is turned into a 3~algebra by setting

(xn 0 • • • 0 xi)(ym 0 • • • 0 3/1) = xn 0 • • • 0 xi 0 ym 0 • • • 0 j/i and m(b) = b e EQ0. As a 3-algebra the full Fock module has a universal property which parallels the universal property of the tensor algebra. See Reference11 for details. With the help of the flip isomorphism it is possible to define permutations on the n-fold tensor product E&n of a centered 3~3~module E in an obvious way. Each permutation is an element of Ba(E°") with the inverse permutation being the adjoint.

211

2.4 Definition. Let E be a centered semi-Hilbert 3-3~rnodule. We define the number operator N in £ a (F 3 (£)) by setting N \ (EQn) = n. The symmetrization operator P in Ba(Fj,(E}} is denned by P \ (E&n) being the mean over all permutations on EQn. We define the symmetric Fock module F3 (E) over E by setting F3 (E) — PF^(E). On F3(.E) we define the creators a+(x) = P^/N(.+ (x) and the annihilators a(x) = i(x)\fNP for all x £ E. Sometimes, we call f+ and (, the free creators and annihilators and a+ and a the symmetric creators and annihilators, respectively. 2.5 Remark. P is a self-adjoint projection. N is self-adjoint. Therefore, a+(z) and a(x) are adjoints. One easily checks PN = NP, a+(x)P = a+(x) and Nl+(x) = l+(x)(N + 1). By these relations the creators and annihilators fulfill a(x)a+(y)-a+(y)a(x)

= (x,y),

(1)

if at least one of the arguments is in the 3~center of E. Clearly, F3(.E) when equipped with the multiplication

(P\fNF}(P\fNG]

= P^/N(FG)

(F, G e F5(E))

is a centered 3~commutative 3~algebra. As a centered 3-commutative 3~ algebra it has a universal property which parallels the universal property of the symmetric tensor algebra. See Reference11 for details. One easily checks the functorial property

by looking at elements of the center. 2.6 Theorem. The * -algebra A(E) of adjointable operators onT^(E) generated by a+(x) (x £ E) and 3 is a centered ^-algebra. Moreover, considering A(E) 0 A(F) as an algebra of operators on r3(£) 0 T 3 (F), we have A(E ® F) = A(E] 0 A(F). This means, in particular, that A(E) and A(F) are Base ^-independent ^-random variables in the centered ^-quantum probability space (A(E © F ), (1, •!}). Proof. Obviously, we have a+(zxz') = za+(x~)z' (x € E; z, z' € 3)- Henceforth, the *-algebra generated by a+(C^(E)) is contained in C$(A(E)) and generating for A(E). This proves the first assertion.

212

The second assertion follows by decomposition into centered elements and by the observation that a+(x) and a(y) commute by (1) for x G

3. Examples Suppose that zi,zz,Z are elements of 3 which fulfill the relation

= 0. Let H be a Hilbert space and E be the centered pre-Hilbert 3-3-module H ® 3 with inner product (/ ® z, f z'} = {/, f)z*z'. Then for x = f ® z\ and y = g ® z2 we find for the 3-random variables a+(:r) and a + (y) in A(E) the relation a(o;)a + (y) - Za+(y}a(x) = ( x , y ) . In the following two examples we choose for 3 the C* -algebra A which underlies Woronowicz's quantum group SUg(2) (q € (—1,1)); see Reference18. This algebra may be considered as the universal enveloping C""-algebra generated by indeterminates a, 7 subject to all relations, which arise by requiring the matrix ( " ~^T j to be unitary. One finds QKyC*) = qj^a, 77* = 7*7, and 1 = a* a + 7*7 = aa* + y. Hence, the whole subspace H ® aj of E classifies a family of random variables, where an arbitrary pair fulfills Z-commutation relations, however, with Z being not a number in C, but, an element of 3 (i.e. a module scalar). Notice also that whenever z\ = z% = z the creators to all elements of the subspace H C3> z commute among themselves. 3.3 Example. The equation z*z — qzz* = 0 admits no (non-trivial) solution z in the bounded operators, unless \q\ = 1 . (If \q\ ^= 1, then \\z\\ ^ ^/\q\ \\z\\, unless | z\\ = 0, i.e. z = 0.)

213 However, for q > 0 there exists an unbounded solution. Consider a pre-Hilbert space h with orthonormal Hamel basis (f-k]k^- Then zek — k

q~iek-i defines a solution. Like indicated in Reference11, we can generalize our notions to preHilbert modules over arbitrary operator *-algebras. Therefore, in this generalized framework there exist families of random variables which fulfill the original ^-commutation relations for q > 0. In Reference3 Bozejko and Speicher introduced a new inner product on the full Fock space which made the creators and their formal adjoints fulfill ^-commutation relations. However, in Reference6 van Leeuwen and Maassen showed that the 'joint distribution' of these 'quantum random variables' is not determined, as it should be, by their separate distributions. The authors of Reference6 conclude from this fact that the convolution on the q-Fock space in Reference3 is not an example for a ^-convolution. Our examples show possibilities of how to write down Bose 3-independent 3-random variables which fulfill ^-commutation releations. However, the relations are due to the choice of elements. The convolution which arises is the generalization of the usual convolution of states to the convolution of 3-states on centered 3~algebras, not a ^-convolution. Recall also that 3-states map into 3- It seems, therefore, difficult to compare our methods with other existing convolutions for states with values in C. Notice, however, that for a 3~quantum probability space (^4, f ) we obtain many states on A by considering ip o 1) 0 (ip o y?2) := V' ° (fi © V^) we may define a 'convolution' of the states i/> o ^ (i = 1,2). We consider it as an interesting question, wether other convolutions can be recovered in this manner. We know that this is possible at least for the central limit distribution of boolean independence; see Reference12.

References 1. L. Accardi and M. Skeide. Interacting Fock space versus full Fock module. Preprint, Rome, 1998. Revised 2000. To appear in Mathematical Physics, Analysis and Geometry. 2. B.V.R. Bhat and M. Skeide. Tensor product systems of Hilbert modules and dilations of completely positive semigroups. Infinite Dimensional Analysis, Quantum Probability & Related Topics, 3:519-575, 2000. 3. M. Bozejko and R. Speicher. An example of generalized Brownian motion. Commun. Math. Phys., 137:519-531, 1991.

214

4. B. Kiimmerer and H. Maassen. The essentially commutative dilations of dynamical semigroups on Mn. Commun. Math. Phys., 109:1-22, 1987. 5. E.G. Lance. Hilbert C*-modules. Cambridge University Press, 1995. 6. H. van Leeuwen and H. Maassen. An obstruction for q-deformation on the convolution product. Preprint, Nijmegen, 1995. 7. M.V. Pimsner. A class of C*-algebras generalizing both Cuntz-Krieger algebras and crossed products by Z. In D.V. Voiculescu, editor, Free probability theory, number 12 in Fields Institute Communications, pages 189-212, 1997. 8. M. Schiirmann. White noise on bialgebras. Number 1544 in Lect. Notes Math. Springer, 1993. 9. M. Schiirmann. Non-commutative probability on algebraic structures. In H. Heyer, editor, Probability measures on groups and related structures XI, pages 332-356. World Sci. Publishing, 1995. 10. M. Skeide. A note on Bose 3~independent random variables fulfilling qcommutation relations. Preprint, Heidelberg, 1996. Submitted to Quantum Probability Communications XL 11. M. Skeide. Hilbert modules in quantum electro dynamics and quantum probability. Commun. Math. Phys., 192:569-604, 1998. 12. M. Skeide. A central limit theorem for Bose 3~independent quantum random variables. Infinite Dimensional Analysis, Quantum Probability & Related Topics, 2:289-299, 1999. 13. M. Skeide. Generalized matrix C*-algebras and representations of Hilbert modules. Mathematical Proceedings of the Royal Irish Academy, 100A:ll-38, 2000. 14. M. Skeide. Hilbert modules and applications in quantum probability. Habilitationsschrift, Cottbus, 2001. Available at http://www.math.tu-cottbus.de/INSTITUT/lswas/.skeide.html. 15. R. Speicher. Combinatorial theory of the free product with amalgamation and operator-valued free probability theory. Number 627 in Memoires of the American Mathematical Society. American Mathematical Society, 1998. 16. D. Voiculescu. Dual algebraic structures on operator algebras related to free products. J. Operator Theory, 17:85-98, 1987. 17. D. Voiculescu. Operations on certain non-commutative operator-valued random variables. Asterisque, 232:243-275, 1995. 18. S.L. Woronowicz. Twisted £77(2) group. An example of a non-commutative differential calculus. Publ. RIMS Kyoto Univ., 23:117-181, 1987.

DILATION THEORY AND CONTINUOUS TENSOR PRODUCT SYSTEMS OF HILBERT MODULES

MICHAEL SKEIDE Lehrstuhl fur Wahrscheinlichkeitstheorie und Statistik, Brandenburgische Technische Universitat Cottbus, Postfach 10 13 44, D-03013 Cottbus, Germany, E-mail: skeide@math. tu-cottbus. de, Homepage: http://www.math.tu-cottbus.de/INSTITUT/lswas/-skeide.html The investigation of products systems of Hilbert modules as introduced by Bhat and Skeide6 has now reached a state where it seems appropriate to give a summary of what we know about the structure. After showing how product systems appear naturally in the theory of dilations of CPsemigroups, it is one of the goals of these notes to give a list of solved and open problems. In contrast with Arveson1, who starts his theory of product systems of Hilbert spaces (Arveson systems, for short) with a concise definition of measurability conditions (which are equivalent to similar continuity conditions), the theory of product systems of Hilbert modules (in the sense of Definition 3.1 below) developed so far works without such conditions. While the algebraic constructions which work in that framework behave nicely with respect to topological completions or closures at a fixed "time", we could show continuity results for time evolutions only in special cases. It is the second goal of these notes to launch a definition of continuous tensor product system (Definition 7.1)(°-a) and to show that this definition, although sufficiently general to contain all reasonable cases, does not have the described defect. In the case of type I and type II systems we found already a way to formulate continuity conditions in a less intrinsic way. Theorem 7.5 shows that the new definition is compatible with these special cases. We are, finally, able to define what we understand by a (continuous) type III product system, thus completing the classification scheme from Bhat and (°-a)This definition was obtained in joint work with B.V.R. Bhat and V. Liebscher within a PPP-project supported by DAAD and DST.

215

216

Skeide6, Barreto, Bhat, Liebscher and Skeide3 and Skeide19. 1. Introduction By dilation many authors understand slightly different things. The denominator common to all the different definitions may be described by the commutative Diagram (1) below. Here B is a unital C*-algebra with a unital completely positive (CP-) semigroup T = (?t) t R and A is another unital C"*-algebra with a semigroup -d — (i?t)t£R of unital endomorphisms, i.e. an .Eo-semigroup. The two are linked together by an embedding (i.e. an injective homomorphism) i: B —> A of B into A and an expectation p: A —> B back to B in such a way that ip = i o p is a conditional expectation onto the range i(B) of i, i.e. p(i(6)ai(6')) = 6p(a)6' for all 6,6' 6 B; a € A(1'a)

B

^— B (1)

The idea of dilation is to understand the dynamics T of B as projection from the dynamics i? of .4. In statistical physics the algebras B and A may be considered as algebras of quantum mechanical observables so that B models the description of a small system embedded into a big one modelled by A. In the classical example B is the algebra of random variables describing a brownian particle moving on a liquid in thermal equilibrium and A is the algebra of random variables describing both the molecules of the liquid and the particle. In both cases we say that the irreversible dynamics of the small system described by completely positive mappings is dilated to a more reversible one on the big system described by unital endomorphisms. (: - b ) Already in Bhat and Skeide6 we showed how to construct from a CPsemigroup T on B, i.e. the upper half of Diagram (1), a product system (l- a >Of course, it is possible to identify B via i as a subalgebra of .4 and to consider just the conditional expectation
217

of Hilbert B-B-modules^1'^ and, in the unital case, how to complete the diagram to a dilation. More precisely, we constructed a dilation on a Hilbert module, i.e. in our case A = !Ba(£1) is the C"*-algebra of all adjointable operators on a Hilbert B-module E and E contains a unit vector £ (i.e. (£,£} = 1) such that p(a) = (£, a£). The situation is illustrated in the following diagram.

B

>B I = «,•€>

(2)

A dilation on a Hilbert module is a quadruple (E, $, i, £) such that Diagram (2) commutes for all t. Actually, the dilation constructed in Reference6 is a weak dilation, i.e. the embedding has the special form i = jo with jo(b) := £b£* where £* is the operator x i—> (£,x).( 1 ' d ' A weak dilation on a Hilbert module is a triple (E, •$, £) such that Diagram (3) commutes for all

B

>B =«,•«>

(3)

#t 16

In Skeide we showed (generalizing Bhat's4 approach to tensor product systems of Hilbert spaces in the sense of Arveson1) how to construct a tensor product system of Hilbert B-B-modu\es from the triple (E, •&,£), i.e. the (1-c)The construction of product systems from CP-semigroups, historically the first, is a special case of the construction starting from so-called CPD-semigroups, which we mention only briefly in Footnote (4.d). ^•^In this case, the family j = (jt)tgR + with jt = i?t ° jo defines a weak Markov flow for T in the sense of Bhat and Parthasarathy5, i.e. j t ( l ) j s + t ( b ) j t ( l ) = jt ° Ts(b). f 1 - 6 'Apparently, our set-up where A = "Za(E) seems to be a restriction of the more general Diagram (1). However, in References16'18 we point out that our notion of dilation is sufficiently wide to contain all explicit and most known abstract examples of dilations in the sense that A. is contained in some "Ba(E) to which the EQ—semigroup •& extends. (For the case of an automorphism white noise the statement follows from Footnote (2.b).) Moreover, all of these known dilations become weak dilations when i (usually unital) is replaced by jo (usually non-unital). For the time being we content ourselves with the knowledge that considering dilations and weak dilations on Hilbert modules is a fairly general frame and the study of these by means of their product systems (for instance, via classification) showed already up to give many new insights.

218

lower half of Diagram (1), at least, when the endomorphisms tit are strict (a condition replacing the normality assumption in the case of Hilbert spaces). It turns out that those Hilbert modules which have a unit vector form an important subclass of the class of all Hilbert modules. It is the goal of these notes to describe the mentioned constructions of product systems in more detail. Where possible we explain the major ideas or even provide short proofs. For didactic reasons we reverse the historic order and start (after repeating in Section 2 some preliminaries about Hilbert modules) in Section 3 with the construction of a product system from a strict E0-semigroup •& on "Sa(E). This allows us to motivate related notions like units and central units for product systems directly from dilation theory. This way to construct product systems is also more directly related to the way how Arveson discovered product systems of Hilbert spaces. Central units are tightly connected with white noises where we call the triple (£,$,£) a white noise, if i? leaves invariant f> = {£,•£), i-e. if p o •dt = p for all t. In other words, (E, fl, £) is a weak dilation of the trivial semigroup.(1-f) We will classify product systems admitting central unit as spatial product systems. Spatiality of product systems of von Neumann modules which have units is equivalent to the results by Christensen and Evans7 on the form of the generator of a normal uniformly continuous CPsemigroup on a von Neumann algebra and, therefore, a deep problem. In Section 4 we define units and central units. Then we set up our classification scheme, which is, like that for Arveson systems, based on units. We repeat simple Examples from Reference3 which show that the refinement of Arveson's classification scheme (in that there are two types of units) and also the distinction into norm and strong topology are really necessary. The particular importance of spatiality we point out in Section 5. We show that a generator of a uniformly continuous CP-semigroup is a Christensen-Evans generator, if the associated GNS-system (see Footnote (4.d)) is spatial (or can be embedded into a spatial one). This allows to identify in the following Section 6 spatial type I systems as time ordered (l-f)'\V e know that it is in some sense provocant to call this a white noise as 'white noise' is something which has to do with 'independence' and, in fact, our white noises come shipped with subalgebras ^4[s,t] C
219

Fock modules. It also shows us that spatial product systems have an index generalizing that of Arveson systems. In the final Section 7 we solve the outstanding problem to define continuous product systems. We show that our definition extends our preliminary definition for type I and type II systems based on the extistence of a continuous unit. The definition is motivated from properties of product systems constructed from strictly continuous .Eo-semigroups and, of course, also such product systems fulfill our definition. Now we, finally, have a chance to solve also the reverse problem, namely, to construct an .Bo-semigroup from a continuous product system (known for non-type III) in full generality. 2. Preliminaries on Hilbert modules and conventions We repeat the basic definitions and constructions for Hilbert modules. For a detailed introduction to Hilbert modules (adapted to our needs) we refer to Skeide18, for a quick reference to Bhat and Skeide6. The book of Lance11 provides a general introduction to Hilbert modules. Throughout these notes A, B,C,... denote unital C""-algebras. 2.1 A pre-Hilbert B-module is a right ^-module E with a sesquilinear inner product (•,•): E x E —> B which is positive ( ( x , x) > 0), right linear ((x,yb) = (x,y)b) and definite ({a:, a;) = 0 => x = 0). If the inner product fails to be definite, then the Cauchy-Schwarz inequality (x,y)(y,x)

< \\(y,y)\\(x,x)

(4)

tells us that we may divide out the submodule KB = {x: (x, x) = 0} of length-zero elements and obtain a pre-Hilbert module. It tells us also that \\x\\ = T/\\(X,X)\\ defines a (semi-)norm. A Hilbert B-module is a pre-Hilbert S-module which is complete in this norm. Every pre-Hilbert S-module E may be completed and we denote the completion (as with all other normed spaces) by E. The isomorphisms among (pre-)Hilbert ^-modules are the unitary (i.e. surjective inner product preserving) mappings.(2-a) A Hilbert A-B-module (or just two-sided Hilbert module) is a Hilbert S-module with a non-degenerate (*-)representation of A by elements in (2-a> Observe that a unitary u has an adjoint, namely, u* = u~1. Therefore, it is right linear and (by isometry) bounded. Adjointable mappings which have a Hilbert module as domain or as range are bounded by the closed graph theorem. For pre-Hilbert modules this need not be so.

220

the (C"*-algebra) S°(£') of adjointable (and, therefore, bounded and right linear) mappings on E. By !Ba'W(.E) we denote the subspace of bilinear or two-sided mappings. In particular, an isomorphism of two-sided Hilbert modules is a two-sided unitary. 2.2 The, by far, most important way how Hilbert modules, in particular, two-sided Hilbert modules appear in dilation theory is the GNS-construction for a completely positive mapping T: A—>B. The GNS-module of T is that Hilbert A-B-modu\e E generated by a vector £ (i.e. E = spar\A£B) and with inner product (a£b,a'£b') = b*T(a*a')b'. This module is determined uniquely by the requirement (£,a£) = T(a)/ 2ib ) 2.3 The tensor product E 0 F of a Hilbert A-B-module E and a Hilbert S-C-module F is the Hilbert ,4-C-module which is the closed linear span of elementary tensors xQy whose inner product is defined by (xQy, x'Qy'} = (y, (x, x'}y'Y (By '®' we always denote the tensor product of vector spaces, usually, completed in some natural norm.) If T, 5 are completely positive mappings A -^-» B -^-> C and if (E, £) and (F, £) denote their GNS-constructions, then S1 o T(a) = {£ 0 C, a£0 C}In other words, the submodule span.4£ 0 £C of E © F with cyclic vector £ 0 C is the GNS-module of S o T. 2.4 If B is a von Neumann algebra acting (non-degenerately) on a Hilbert space G, then G is a Hilbert S-C-module. For some Hilbert B-module E we construct the Hilbert space H — E 0 G. For every x e E we define the mapping Lx: g H-*• x 0 g in Ti(G,H) whose adjoint is determined by LX'-V 0 5 >-* ( x , y ) g . Moreover, L*xLy = (a;, j/} so that we may identify E1 (isometrically) as a subset of ^{G^H}. We call the mapping r):xt-^Lx the Stinespring representation of £. Following Skeide17, we say E is a von Neumann B-module, if it is strongly closed in CB(G, H). (2.b)jf j js an embedding and p an expectation as required in a dilation, then we have even E = A£ and i(fe)£ = £6. (This situation is most similar to usual GNS-construction for a state.) What happens, if there is an automorphism a of A leaving p invariant, i.e. p o a = p? Then two short computations show that the mapping u: a£ —> a(a)£ defines a unitary on E such that a(a)x = uau"x for all x 6 E. If the GNS-representation is faithful so that we may identify A as a subset of "Za(E~), then a(a) = uau" and the automorphism a extends to a unitarily implemented automorphism of "Ba(E). If the GNS-representation is not faithful, then the computations show that a repects the kernel of the GNS-representation so that we may divide out this kernel. Therefore, as soon as we are concerned with a white noise of automorphisms, we may divide out the kernel of the expectation p and pass to a (unitarily implemented) white noise on a Hilbert module.

221

In contrast with general Hilbert modules, von Neumann modules are always self-dual, i.e. for every bounded right linear mapping <5: E —> B there exists a (unique) element x e E generating $ as $(y) = x*y := (x,y). Like for Hilbert spaces one shows that bounded right linear operators on (or between) von Neumann modules are adjointable. Self-duality also guarantees that for any strongly closed submodule F C E there is a (unique) projection p e "Ba(E) onto F. Also this is a fact that need not be true for general Hilbert modules. If E is a Hilbert .4-S-module, then p(a)(x 0 g) = (ax) 0 g defines a representation p: A —> 'B(H) which we call the Stinespring representation of A associated with E. (In particular, i f A = B a (£), then p identifies ^(E) as a subalgebra of "S>(H), even a von Neuman algebra on H, if E is a von Neumann module.) We say E is a von Neumann A-B-module (or a twosided von Neumann module), if it is a von Neumann S-module and if the Stinespring representation of A is normal. The strong closure E 0s F of the tensor product of two-sided von Neumann modules is again a two-sided von Neumann module. If E is the GNS-module of a normal completely positive mapping T between von Neumann algebras A and B C 23(G), then the strong closure E3 C S(G,/f) of £ is a von Neumann A-B-module. Moreover, p: A —» 'B(H) is, indeed, the original Stinespring representation of A and the mapping L^ for the cyclic vector £ e E fulfills T(o) = L^p(a)L^S2-c^ 2.5 By far, the most concrete results in dilation theory are obtained for the von Neumann algebra B = 23 (G) and the dilations act on the algebra 'B(H) where the Hilbert space H usually has the form H = G (g) ft for some other Hilbert space F). Why is this so? The answer lies in the simple structure of von Neumann 3(G)-modules and, in particular, of von Neumann 3(G)23(G)-modules. Since S(G) contains the finite-rank operators, the von Neumann S(G)-module E C "S(G,H) contains the finite-rank operators of 3(G, H) (or, to be more precise, at least those to elements in the total subset EG of H) and, because E is strongly closed, we find E = ®(G, H). One easily checks that also 3°(E) = 'B(H). Therefore, dilations of CPsemigroups on a von Neumann 23(G)-module act on ( 2 - c ) Notice that the Stinespring representations of two completely positive mappings T, S do not help us in recovering the Stinespring representation of SoT. On the contrary, the GNS-modules, being functors which send representations of the algebra to the right to representations of the algebra to the left, compose under tensor product to the functor for the composed mapping SoT; cf. Section 2.3.

222

Moreover, if E is a two-sided von Neumann S(G)-module, then the representation p of "S(G) on H is normal (and non-degenerate). Therefore, H — G ® ft for a suitable Hilbert space ^ and p — Ida 1 in that identification. This explains clearly why problems which, for general C*-algebras, require Hilbert module methods can be solved on G ®ft in the case 23 (G). Can we specify ft further? The answer is simple. The elements h of ft are in one-to-one correspondence with those mappings idc<8>h 6 23(G, G®ft) which commute with all elements in 2.6 The strict topology of 23a(.E) arises by the observation due to Kasparov9 that !B°(£) is the multiplier algebra of the C*-subalgebra of compact operators OC(E) which is generated by the rank-one operators xy":z ^ x(y, z ) . In other words, ®a(£l) is the completion of DC(£) in the topology generated by the two families a i-» ||a&|| and a i-> ||fca|| (k e OC(E)) of seminorms. Here we follow Lance' convention11 and by the strict topology we mean always the restriction to bounded subsets of 23a(.E). One can show that on the ball the strict topology coincides with the *-strong topology. In the case of Hilbert spaces the strict topology is the *-cr-strong topology. It is well-known that normal representations of 'B(H) are also *cr-strong, so for Hilbert modules the strict topology on the ball is, indeed, an appropriate substitute of the normal topology. (2'e) We used already the well-known fact that normal (non-degenerate) repesentations of "B(G) on H decompose H as G ft. Can we do the same for Hilbert modules? The answer is yes, if the (non-degenerate) representation i9:Sa(£) -> ^(F) is strict (cf. also Footnote (2.e)), and if E contains a unit vector £.' 2 ' f ) Here we restrict to the case F = E, i.e. i? is a strict unital endomorphism of 3 a (E).( 2 - g > ( 2 - d >Defming the center of a two-sided S-module E as CB(E) = {z e E: bx = xb (6 6 B)}, we notice that for central elements xi, X2 the inner product takes values in the center of B. KB — S(G), then the center of B is trivial so that the inner product of central elements in E is a scalar multiple of 1. Identifying this scalar with the inner product of the corresponding (unique) elements of /i, /2 £ Sj such that Xi ~ idc ® fi gives us the isomorphism. (2'e'ln the sequel, what we need from the strict topology is the fact that a (bounded) approximate unit for the compact operators, or even the finite-rank operators ff(E) := spanEE*, converges to id^. Any other topology which also has this property will also serve this purpose and may replace the strict topology. ( 2 - f )Xhis result and others which are valid only under the assumption of a unit vector make this property so important that we give a name to it, and call E a unital Hilbert module. ( 2 -s)\Ve discuss in Skeide16 that the result is valid for arbitrary Hilbert modules F, even

223

So let E be a unital Hilbert 6-module and i? a strict unital endomorphism ofBa(E). Put E% = $(££*)£ and define a (unital) left multiplication on this Hilbert submodule of E by bx = i?(£&£*)a;. One easily shows that the mapping u:xQy i—> i9(i£*)y defines an isometry E 0 E$ —* E. Using an approximate unit for J(E) and strictness of $ one shows that u is surjective, hence, unitary/2-11' In the identification E = E 0 E( we find that i?(a) € S°(£l) acts as i9(a) =

3. From £70-semigroups to product systems of Hilbert modules Let (E, $, £) be a unital Hilbert B-module E 9 £ and and let i? be a strict .Eo-semigroup on 3a(E). A simultaneous application of our representation theory in Section 2.6 to all $t provides us with a family EQ = (Et)t€R of Hilbert S-B-modules Et = ptE (pt := $t(££*)> left multiplications bxt = tit(£bg*)xt) and unitaries ut:E 0 E4 defined by ut(x 0 y t ) = i?t(^*)j/tObserve that EQ = B as two-sided module via £ i—> 1. We define the restrictions ust = "t t (Es O Et)- These map into Ea+t, because ps+ttft(zs£*)yt = dt(psXs£*)yt = flt(xa£*)yt- On the other hand, they are onto Ea+t- (Write z £ Ea+t c E = ut(EQEt) as z = Since ps+tz = z we have

£ u3t(Es 0 £t)-) Therefore, uat:Es Q Et —> Es+t are unitaries. Moreover, from ust(bxaQyt) = tft(tfs(£b£*)xs£*)yt = bust(xsQyt) we see that the uat are two-sided. Finally, we easilly verify the associativity conditions

0u s t) =ut(uaQ\dEt)

(5)

0 uat) = u(r+s}t(ura 0 id £t ).

(6)

over a different C*-algebra C. ' 'In both computations it is an important step to insert a one in the rank-one operator xy* = xly* = x£*£y* and to use then the *-homomorphism property of i9, when applied to that product of operators x£* and fj/*. ( 2 -')One can show that up to two-sided isomorphism E$ does not depend on the choice of £. If -9, •&' are two strict unital endomorphisms of B°(B) than E^ and E', are isomorphic, if and only if •& and •&' are conjugate, i.e. if i9' = utfu* for some unitary u 6 23" (ZJ).

224

We observe that uot: b 0 xt H-> bxt and w to : xt 0 6 i-> xtb give us back the canonical identifications B 0 Et = Et = Et 0 B. Collecting the majority of these results we see that EQ is a, product system in the sense of the following definition from Bhat and Skeide6. 3.1 Definition. A product system of Hilbert modules is a family EQ = (Et) t£R of Hilbert #-S-modules Et (E0 = B) with a family ust of unitaries in "Ba'M(Es 0 Et,Es+t] fulfilling the associativity condition (6) (u0t,uto being the canonical identifications). (3-a) Once the mappings ust (and in the case of the preceding product system coming from an jEo~semigroup the mappings ut) are fixed, we use the identifications Es®Et = Es+t (&ndEoEt = E).

(7)

(Obviously, if EQ is the product system constructed from an .Eo-semigroup i?, then we recover fl in this identification as $ t (a) = a 0 id# t .) Several natural questions arise. 3.2 Question. What is the connection with Arveson's product systems of Hilbert spaces (Arveson systems for short) which start from normal EQsemigroups on S(G)? Our construction (including the representation theory for 25°(.E)) is a direct generalization from Bhat's4 approach to Arveson systems. (3'b) Arveson requires additional measurability conditions on a product system which are fulfilled, if we start with an -E0-semigroup which is continuous (pointwise on 23 (G)) in the strong operator topology. Additionally, the Hilbert spaces of an Arveson system are all isomorphic (3-a'This is the definition of product systems of Hilbert modules. It has an obvious version for von Neumann modules, where all appearing (operator) spaces should be strongly closed. However, we do not intend to go too much into the technicalities of von Neumann modules. It was our intention to give a precise definition in Section 2.4, because some of our classification results hold only for von Neumann modules, and because of the importance of the case B = B(G) in Section 2.5 when we want to compare with existing results. Let us mention, however, that, starting from an £7o—semigroup i? on "Ba(E) where E is some unital von Neumann module, we may construct a product system of von Neumann modules as before, provided that all $t fulfill the weaker condition to be normal mappings. ( 3 - b ) Arveson's1 approach is based on Footnote (2.d) and relies on the simple structure of ®(G)-modules. It does not generalize to the module case. When we construct as in Example 4.5 the product system E® of two-sided von Neumann ®(G)-modules for an Bo-semigroup on B(G), then the Hilbert spaces fit such that Et = 25(C, G ® fit) form the corresponding Arveson system.

225

(infinite-dimensional separable). We will see in Example 4.7 that we cannot hold this condition. In Section 7 we propose a suitable definition of continuous tensor product systems. 3.3 Question. Do we obtain all product systems by the preceding construction?^3-^ This question is closely related to the correct notion of measurability. It can be answered in the affirmative sense for Arveson systems; see Reference2. Certainly our answer to the measurability problem should be judged by checking whether it allows to preserve Arveson's result that all Arveson systems arise from .Eo~semigroups on ^(G) also in the case of product systems of Hilbert modules. We are not yet able to solve that problem, however, we show at least that our definition of continuous product system is not too restrictive. 3.4 Question. How can we classify product systems? Like for Arveson systems our classification scheme is based on how many units we have; see Section 4. However, it turns out that we have to distinguish between general units and central units/3'*1) This leads to a refined classification as compared with that of Arveson systems. However, we are able to present simple examples (even of type III systems — a difficult issue for Arveson systems) which show that our refinement is necessary. Of course, our classification is made in such a way that most results by Arveson show to remain true also for product systems of Hilbert modules. In the first place, we are able to preserve the distinguished role played by the symmetric Fock space which becomes now the time ordered Fock module. 4. Units in products systems Arveson systems, so far, are classified by their units (families of vectors in the members of the Arveson system which factorize into elementary tensors in a stationary way). The basic example of an Arveson system is the family T®(K) = (rt(A")X€R (K some Hilbert space) of boson Fock spaces Tt(K) = r ( L 2 ( [ 0 , t ] , K ) ) which factorize as T,(K)®rt(K)

* r(L2((t,t + s ] , K ) ) ® T t ( K )

* Ts+t(K).

(3'c'The second construction of product systems starting from CP- or CPD-semigroups (cf. Footnote (4.d)) is even less exhaustive as it leads necessarily to type I systems (cf. Footnote (4.f)). ( 3 - d >Central units for a product system of von Neumann ®(G)-modules correspond precisely to units for the central Arveson subsystem as discussed in Footnote (3.b).

226

The units have the form ut = e^C^t]/) (c £ C, / e K) where denotes the exponential vector to x £ L?([Q, t ] , K ) .

The Arveson system T®(K) is generated by its units (there is no proper subsystem containing all the units). Such Arveson systems are said to be type I and Arveson showed that all type I Arveson systems have the form T®(K) for a (unique up to isomorphism, i.e. up to dimension) Hilbert space K. The dimension of K, called index, is a complete isomorphism invariant of type I Arveson systems. An Arveson system is type II, if it has a unit, but is not type I. It contains a unique maximal type I subsystem and the index of a type II system is that of its maximal type I subsystem.^4-3-) Recent work of Tsirelson20 and its systematic extension by Liebscher12 show that there is an abundance of (mutually non-isomorphic) type II systems having the same index. So the index is certainly not a complete isomorphism invariant for type II systems. Finally, an Arveson systems is type III, if it has no units. Existence of type III Arveson systems is known since ever, but also here only recently Tsirelson21 has constructed an explicit example. Unlike, for Arveson systems, where the notion of unit is put into evidence by the importance of the results derived from it, for product systems of Hilbert modules we have a possibility to motivate this notion. Let us recall that we are particularly interested in the case when the (strict) EQsemigroup $ is a (weak) dilation of some (unital) CP-semigroup, or even a white noise. We repeat a result from Skeide16. 4.1 Proposition. For the triple (E,tf,£) equivalent.

the following conditions are

(1) The family pt = i?t(££*) of projections is increasing, i.e. pt > po for all t € T. (2) The mappings Tt(b) = {£, $*(£&£*)£) define a unital CP-semigroup T, i.e. (E,&,£) is a weak dilation. (3) Tt(l) = I for all t £ T. Under any of these conditions the elements £t = £ 6 Et C E fulfill

6 © 6 = 6+t,

(8)

( 4 - a * Indices behave additive under tensor product of Arveson systems, thus, justifying the name 'index'.

227

£o = 1 and Tt(b) = (&,&&)• Moreover, T is the trivial semigroup, i.e. (E,$,£) is a (weak) white noise, if and only if all £t commute with all b&B. This encourages the following Definition from Bhat and Skeide6. 4.2 Definition. A unit for a product system EQ is a family £0 = (6) t€K of vectors £t € Et fulfilling (8) and £0 = 1- The unit £0 is unital, if it consists of unit vectors. It is central, if all & commute with all b 6 B. By we denote the set of all units So far, this is a purely algebraic definition. Even the case that a unit is 0 except at t = 0 is allowed. Units in an Arveson system must satisfy certain measurability conditions. These conditions imply, in particular, that for any two (non-zero) units u®,u'® the mapping 1 1-> (ut,u't), which obviously is a semigroup in C, is measurable, hence, continuous. ( 4 - b ) In our frame it turns out that we obtain the most satsifactory results, if we base our classification on continuous sets of units. (4'c) Like for Arveson units it turns out that matrix elements of units have a semigroup property. However, as we learned already in 2.3, instead of looking just at matrix elements {&,£*) we nave to switch our interest to (bounded) mappings 6 i-» {&,&£«) which, clearly, form a semigroup on B. The collection of all these semigroups fulfills a positivity condition, namely, it is a completely positive definite (CPD) kernel in the sense of References3'18.(4-d) 4.3 Definition. By it = (Ut) t€K , where the completely positive definite ' 'The covaricmce function defined on the pairs of all measurable Arveson units as the derivative of (ut,u't) at t = 0 is a conditionally positive definite kernel. From here it is quite easy to show that an Arveson system generated by its units consists of symmetric Fock spaces. (4-c)\Ve are speaking about norm continuity. It is an open problem to decide, whether this may be weakened to norm measurability. Also weaker topologies coming from weaker topologies on B are thinkable. However, Example 4.7, which has only strongly continuous units, tells us that we may not expect to derive similar results for weaker topologies. ' 'We show also that every CPD-semigroup, continuous or not, arises in this way from matrix elements of units in a product system. The construction of that product system is very much like a GNS-construction and, therefore, we call it the GNS-system of the CPD-semigroup. The first construction of a product system from a CP-semigroup in Reference6 appears as a special case of the GNS-construction for CPD-semigroups.

228

kernel iit:U(E&) x U(EQ) -* S(B) is defined by

we denote the CPD-semigroup associated with £0. A set 5 C U(E&) of units is continuous, if the CPD-semigroup il \ S is uniformly continuous, i.e. if the semigroup Uf'? is uniformly continuous for all £®,£' 0 € S. In particular, a single unit £0 is continuous, if the set Now we are ready to set up our classification scheme. 4.4 Definition. A product systems of Hilbert modules E& is type I, if there is a continuous set 5 of units which generates E® (i.e. E® is the smallest subsystem of E® containing all units of S) . E® is type II, if it has a continuous unit, but is not type I. It is type III if it has no continuous unit.( 4 - f ) A product system is spatial, if it has a central unital unit u>®, and completely spatial, if it is also type I and the generating subset S can be chosen to contain w®.( 4-g ) Clearly, a central unital unit is continuous .({wt,«w t ) = idg is constant and, therefore, continuous). So, a spatial product system is clearly nontype III/4'h' One main result of Reference3 asserts that non-type III product systems of von Neumann modules are spatial, automatically. We can determine completely the form of completely spatial systems and, therefore, also of type I systems of von Neumann modules: They are all (systems of) time ordered Fock modules; see Section 6. In Example 4.6 we describe a type I product system without central unit. This shows us that non-type (4.e)Qne may show that for checking continuity of 5 it is sufficient that E contains one continuous unit f® and that the matrix elements (ft, £'t) and (£'t, £'t) depend continuously (in B) on t for all f' 0 6 S; see References3'18. (4-f)The GNS-system of a CPD-semigroup (cf. Footnote (4.d)) is, by definition, generated by its units. Therefore, if the CPD-semigroup is uniformly continuous, then the GNSsystem is type I, automatically. (4-g>We are speaking about Hilbert modules. The preceding definition has analogues for algebraic product systems of pre-Hilbert modules (with types denoted by I, and so on) and for product systems of von Neumann modules (with types denoted by I s , and so on. The continuity required for the units is, however, the same in all cases. (4.h)gy Reference3 a continuous unit may be normalized to consist of unit vectors. Therefore, in the definition of spatial we may replace central unital unit by central continuous unit.

229

III product systems of Hilbert modules need not be spatial and that type I product systems need not be time ordered Fock modules. On the other hand, one of the main results of Reference3 asserts that non-type III product systems of von Neumann modules are always spatial. Therefore, type I products systems of von Neumann modules are always time ordered Fock modules. Example 4.7 shows us that it is easy to write down product systems even of von Neumann modules which have not a single continuous unit. Nevertheless, this product system is generated by a single strongly continuous unit. This shows that classifications based on units which are continuous in a weaker topology only may be quite different. As a typical feature we find that, in particular, commutative algebras (in contrast with the extremely non-commutative 23(G)) provide us with interesting counter examples. 4.5 Example3. Let B denote a unital G*-algebra. Then B is itself a Hilbert S-module (with inner product (b,b') = b*b') with unit vector 1 and 1!>a(B) — B. Let i? be an £b-semigroup on B. Then the associated product system is Et — B as Hilbert H-module, but with left multiplication b.xt = tit(b)xt- Clearly, EQ has a unital unit £0 with £t = 1. This unit is continuous, if and only $ is uniformly continuous.(4'') In particular, if i?t(&) = utbu* is a (semi-)group of inner automorphisms (for some unitary group u in B), then ut:B —> Et,x i—> utx establishes an isomorphism from the trivial product system (^) teK to E®. One can show that the product system E® is isomorphic to the trivial product system, if and only if the automorphisms i?t are inner. Therefore, the fact that automorphism semigroups on 23 (G) have trivial Arveson systems is entirely due to the fact that 3(G) admits only inner automorphisms. Product systems of non-inner automorphism semigroups have interesting product systems and should not be excluded. 4.6 Example3. Let B = 3C(G) + Cl C £(G) be the unitization of the compact operators on some infinite-dimensional Hilbert space G. Let eith be a unitary group on G for some self-adjoint element h G S(G), For the automorphism semigroup $t = elth»e~*th we construct the product systems EQ as in Example 4.5. ' 'This shows clearly that for checking continuity of a unit it is not sufficient to look only at matrix elements (£t,£t); cf. Footnote (4.e).

230

Suppose ijJt € Et is a central element, i.e. elthbe lthwt = u>tb or be~lthujt = e~lthujtb for all b € B. In other words, e~tthut is in the center of B and, therefore, a scalar multiple ctl of 1 so that u>t = ctelth. If ct ^ 0, then it is not difficult to see that the requirment elth e B puts very severe restrictions on h. For a single time t we cannot exclude completely that h £ B. However, if ut is a whole family of central elements in Et (for instance, a unit) with ct ^ 0, then differentiating ^ at t = 0 tells us that h must be in B (as norm limit of elements in B). Consequently, if h £ B, then EQ has no central continuous units, although it is generated by the single continuous unit £t = 1. In accordance with our result that type I systems of von Neumann modules are spatial, the problem dissappears, if we pass to the strong closure 'B(G) of B. 4.7 Example3. Let B = C 0 (R) + Cl C C 6 (R) the unitization of the continuous functions on R vanishing at infinity. On B we define the time shift automorphism (semi-)group s = (st) t6R by setting $tf(s) = f ( s — t). Clearly, §t is not uniformly continuous. We construct the product system EQ as in Example 4.5. Suppose now that £0 is a unit. Then (£t,/£t) = (£t>6) s t/- Suppose £Q was continuous. Then f-stf = «6,/6> - (&,/&)) + ({6,6} - (&>,£o))st/ implies, a contradiciton, that St is norm continuous. Therefore, E® does not have continuous units. This example can be extended in two directions. Firstly, we may restrict to Co(R_)+Cl so that JR_St defines a proper .Eo-semigroup. Secondly, we may pass to the strong closures L°°(R) and L°°(R_), respectively, providing us with analogue examples of product systems of von Neumann modules. The structure of type I systems is remarkably invariant under the choice of the generating continuous subset S of units. In type II systems (or spatial systems) we have to fix a continuous (central) reference unit and in how far there are other units extending the reference unit to a continuous set of units may depend on the choice of the reference unit. The question in how far the continuity (or measurability) structure on the product system depends on the choice of the unit is an open problem even for Arveson systems. Therefore, if we speak about type II product systems, we often include the reference unit into the definition a speak of pairs (Ee, £ 0 ). We discuss in Section 7 the relation to a definition of continuous tensor product system.

231

5. The CPD-semigroup of spatial product systems We mentioned that the crucial point in our classification of type I systems is to establish spatiality (where possible, of course). Let us see why this is so important. So let w® be a central unit for a product system E& and let £® be any other unit. Then Uf"(&) = (&,&u,t) - &,w t >& = 4'w(l)6

(9)

and In other words, 11^^(1) is a semigroup in B and determines V£'w by (9). In particular, ilu''a'(l) is a semigroup in Ce(B). If w0 is continuous, then all il"'w(l) are invertible. Henceforth, we may assume without loss of generality that w® is unital, i.e. H"1" = id is the trivial semigroup. 5.1 Lemma. Let w® be a central unital unit and let £® be another unit for a product system EQ such that the CPD-semigroup it f {u'®,£0} is uniformly continuous. Let /3 denote the generator of the semigroup 11^^(1) in B, i.e. 11^(1) = e^, and let & denote the generator of the CP-semigroup il^ on B. Then the mapping b K^ £«(&)-6/3-/3*6

(10)

is completely positive, i.e. & is a CE-generatorS5-^ PROOF. Since it is a CPD-semigroup, the semigroup il(2) = 042))(eK (2}

/ilu''t*; li1*1'^ ^

M^B) with itj

= f '{|U

^ |4 j is competely positive. Its generator is

r»(2) / " l l

_

f^t'

V6

^12\

on

d

6 / ~" ~di

(^ll) i^t ' (^12) \

(

0

&12/3

\ ttt'^lh \ l[t't(h \ I ~ \ fi*b Ct(h

As generator of a CP-semigroup £^2^ is conditionally completely positive. Let Ai = ( ° M and J3, = ( ° ~ f c i ) . Then AiBi — 0, i.e. ]T AiBi = 0, so V • •/ \ - / 4

that 0

^^

-

z_^ i v «*„*„.

ref.^^.^

(5-a>Christensen and Evans7 established that every generator of a unifomly continuous normal CP-semigroup on a von Neumann algebra decomposes into a completely positive part and a part 6 >-» b/3 + /3* 6.

232

to) - a?a,-/J - /J'aja^ This means that (10) is completely positive. • It is not difficult to prove the multi index version for CPD-semigroups; see References3'18. 5.2 Theorem. Let EQ be a product system with a subset S C U(Ee) of units and a central (unital) unit u>® such that ii \ SU {u>®} is a uniformly continuous CPD-semigroup and denote by £ = J^| t _ 0 iU \ S the generator o/il f S. Then there exists a mapping S —> B, £0 H-> fa such that the kernel

is completely positive definite. In other words, doing the Kolmogorov decomposition for £o we find a Hilbert B-B-module F and a mapping S -» F, £0 i-» ^ such that £ e '£'(6) = (C(,b^i)+bfa>+/3^b.

(11)

VKe say S, has CE-form. 6. The time ordered Fock module and its CPD-semigroup Let F be a Hilbert B-B-module F. Then L 2 (R+, F) is defined as norm completion of the space of F-valued step functions with inner product (x, y) = f(x(t),y(t))dt. Higher tensor powers fulfill L 2 (R + ,F) 0 " = The time ordered Fock module is defined as

n=0

where An is the indicator function of the set {tn > . . . > ti > 0} C M" which acts as projection in the obvious way. By Reference6 the family T O (F) = 0Tt(.F)) t6R of restrictions F t (F) of F(F) to the interval [0, t] forms a product system via the identification

[X.QYt]

=

(stXs}Q[Yt}

where [X] means the function obtained by pointwise "evaluation" of the element X e F(F). Liebscher and Skeide13 show that the set of continuous units (with the vacuum unit w 0 = (w t ), wt = 1 £ I/ 2 (E+,F) G ° = B as

233 00

reference unit) consists of the units £0(/3, C) = (£t)teR teR

+

with £t = ri=0

defined by $ = et(s and where the parametrization by pairs (/?, C) € B x F is one-to-one. generator of the associated CPD-semigroup is

The

In other words, it is a CE-generator. 6.1 Corollary. Let E® be a completely spatial product system with a generating set S C U(EQ) of continuous units. By Theorem 5.2 the CPDsemigroup il \ S has the CE-generator (11) so that the the mapping

£0 ^ e0(ft,c«)

determines an isometric embedding EQ —> IT®(.F). A result from Skeide19 asserts that F can be chosen minimal and that in this case the embedding is onto, i.e. an isomorphism. In other words, completely spatial product systems are time ordered Fock modules. The module F is a complete isomorphism invariant. A further result from Reference19 asserts that every spatial product system contains a unique maximal completely spatial subsystem. 6.2 Definition. The index of a spatial product system is the module F such that the maximal completely spatial subsystem is isomorphic to FQ(F). Reference19 provides also a product of product systems under which the index behaves additive (direct sum), thus, justifying the name 'index'. In the case of Arveson systems this product gives us back the tensor product, if at least one factor is type I. In general, it gives only a subsystem of the tensor product. We mentioned already the result from Reference3 which states that nontype III systems of von Neumann modules are spatial. Thus, type I systems of von Neumann modules are completely spatial and, therefore, isomorphic to (strong closures of) time ordered Fock modules. It is not possible to explain the proof of this result in a few words. We mention that it requires a complete understanding of the endomorphisms of a time ordered product system. The result shows then to be equivalent to the result by Christensen and Evans7.

234

7. Continuous product systems So far, we know completely the structure of completely spatial systems (i.e. also of type I systems of von Neumann modules) and we have simple examples of other types which show that the refinement of our classification scheme, as compared with that of Arveson, is necessary. Applying a technique from Liebscher12, which associates with each Arveson system one-to-one a type II Arveson system, we should be able to produce also lots of type II systems of Hilbert modules. A result missing so far, is that any Arveson system comes from an E0semigroup. a-Weakly continuous normal E^-semigroups on !B(G) for an infinite-dimensional separable Hilbert space G are classified one-to-one up to cocycle conjugacy by product systems. We see that this result depends on both sides on technical conditions: On the semigroup side a(E) for some unital Hilbert S-module E, then we are certainly not comitting an error requiring these conditions. Looking, in how far the conditions may be weakened (for instance, measurability instead of continuity) is a different problem. We will also not show that our conditions will be sufficient to assure that every such product system comes from an .Eo~semigroup(7'a\ These questions we leave for future work. So let i9 be a strict and strictly continuous E0-semigroup on a Hilbert B-module E with unit vector £ and associated product system E®. A first observation is that every member Et of the product system is contained as a right submodule in E. Arveson requires that all members of an Arveson system (for t > 0) are isomorphic (infinite-dimensional and separable). This is true also for the members of a time ordered system which are isomorphic (for t > 0) as Hilbert bimodules. However, Example 4.7 shows that it need not be so. (All members of this product system are isomorphic as right modules but not two different of them are isomorphic as bimodules.) Consequently, we will require that all members of a product system may be embedded as right Hilbert modules into a fixed one.

(7.=

) Although we, certainly, believe that this should be the case.

235

Let us fix x £ E and consider the family (a^) R with xt = ptx. Since i? is strictly continuous, the function t i-> xt e £ is norm continuous. Moreover, E10 is generated by such sections. (Each yt e Et can be written as ptx for a suitable a; e -E.) Furthermore, if (x t ) teR , (j/t) teK (%t,yt £ Et) are continuous families in -E, then the function (s, t) i—> xs 0 3/4 is also continuous. (We have 0 Vt+e - Xs © 3/t

= tit+e(

=tft+e(xs+se)(yt+e- y

which is small, if (8, e) is small in R 2 .) We obtain the following definition by passing from the concrete identification Et C E to a more arbitrary one it'.Et —» E, and expressing the preceding properties in terms of it. 7. 1 Definition. Let E& be a product system of Hilbert B-B-modu\es with a family i = (it) t6K of isometric embeddings it'.Et —> E into a unital Hilbert ,8-module. Denote by CSi(EQ) = <x = (xt)te& • xt G Et, t

is continuous I

the set of continuous sections of E0 (with respect to i). We say .E0 is continuous (with respect to i), if the following conditions are satisfied. (1) For every yt £ Et we can find a continuous section x 6 CSi(E®} such that j/t = Xt. (2) For every pair x, y £ CSi(EQ) of continuous sections the function

is continuous. We say two embeddings i and i' have the same continuous structure, if

7.2 Example. By construction, every product system coming from a strictly continuous strict E0~semigroup •& on some unital Hilbert 5-module is a continuous product system. In particular, if $ is an -Go-semigroup on

236

B, then the product system as constructed in Example 4.5 is continuous, provided that i) is strictly continuous (as, for instance, in Example 4.6). However, since !Ba(B) = B is unital so that the strict topology coincides with the norm topology, the E0-semigroup •d = s on B = C0(R) + Cl as in Example 4.7 is not strictly continuous. Nevertheless, the associated product system is continuous. (The continuous sections are just the continuous functions x:t H-> xt & B where B = Bt as right module. Since functions in Co(R) are uniformly continuous, the semigroup s is Co-continuous, i.e. t H-> §t(b) is continuous for all b e B. Therefore, also the functions ( s , t ) i-> %sG>yt = St(xs)yt are continuous for all continuous sections x, y.) Evidently, this remains true for every product system associated with a Co-continuous £t>-semigroup on a unital C"*-algebra B. By Definition 7.1 the mappings t H-> (xt,yt) are continuous for all x, y e CSi(EQ). By Property (1) for every b e B there exists a continuous section x such that XQ = b. By Property (2) for every y G CSi(EQ) the mapping t H-* (0, t) H-> 2:0 0 yt = &yt is continuous. It follows that also the section (fo/t) t6K is in CSi(EQ). In other words, in a continuous product system the mappings 1 1—» (:rt, »j/t) are Co-continuous for all x, y 6 CSi(E®). A natural question is, whether the units among the continuous sections are continuous in the sense of Definition 4.3. Let us recall the result from References3'18 which asserts that a set of units is continuous, if at least one unit £0 of them is continuous, and if the matrix elements (£t,£t) and (£t,£t) for all other units £'0 depend continuously on t.(7-b} The latter condition is clearly fulfilled for an arbitrary set of units which are continuous sections. It turns out that the same remains true for continuous sections, but, before we prove that we give a concise definition. 7.3 Definition. A continuous product system is uniformly continuous, if for all its continuous sections x,y E CSi(E&] the mapping t i-» (xt, »yt) is uniformly continuous (i.e. continuous as mapping K + Before we show in full generality that a continuous product system with a continuous unit among the continuous sections is uniformly continuous, we consider a special case (which is a slight generalization of the mentioned result from References3'18). (7.b)^s soon gg we have a central a;® unit among the continuous sections we are on the save side, because (ujt,i^t) is then continuous so that the CP-semigroup 6 >-+ {u>t,fcu;t} = (<jjt,ut)b is uniformly continuous.

237

7.4 Proposition. Let EQ be the continuous product system coming from a strictly continuous strict dilation (E, i9, £) and let £0 = (£t) be the unit £t — £ £ Et C E. Then EQ is uniformly continuous, if and only if £0 is continuous, i.e. if and only if the dilated unital CP-semigroup Tt = {&, »£t) is uniformly continuous. PROOF. & embedded into E is constant, so £0 6 CSi(EG). If £0 is not continuous, then £Q is a continuous section whose matrix elements Tt are not uniformly continuous, so neither is E®. To show the converse, let £0 be continuous and let x,y € CSt(EQ). Observe that £ = £ © £t in the factorization E = E 0 Et. We find , byt+£) - (xt, byt) = (xt+£, byt+e) - (£e 0 xt, £e 0 byt)

The norm of xt+£ — ££O:ct is small for e sufficiently small, because £0 and x are continuous sections, and similarly for yt+e — £e O Vt- Consequently, the norm of the mapping which maps 6 to the first plus the second summand is small. The norm of the mapping which maps b to the third summand is small, because ||h& -&6||2 - T£(b*V) - T£(b*)b - b*Te(b) + b*b is ||6||2 times a small number. This is uniform left continuity. To see uniform right continuity replace t by t — e. • It remains, to show that an arbitrary continuous product system with a continuous unit £® can be obtained, including its continuous structure, from a strictly continuous strict dilation. In a certain sense, we have to reverse the construction of a product system EQ from an £fl~semigroup on a Hilbert module E. Fortunately, we have availabe the contructuction from Bhat and Skeide6, which we describe briefly. Suppose the product system has a unital unit £0. Then the mappings "f(s+t)t'-xt ^ £s 0 %t provide us with an inductive system of isometric embeddings Et —> Es+t giving rise to an inductive limit ECO which is a right Hilbert module. Under the canonical embeddings Et —>• -Soo all £t are mapped to the same unit vector £ 6 EX,. We verify EOO O Et — E^ and the associativity condition in Equation (7). Therefore, (£00,1?, £) with i?t(a) = a Q ids, is a strict dilation. Obviously, the product system of (.E^i?, £) is E0 and the unit (OteR §ives us back the unit we started with. If the pair (E 0 ,£ 0 ) comes from a dilation, then (E, £) and (E<x> , £) are canonically isomorphic, if arid

238

only if lim ptE = E. Such dilations are called primaryS7^ Clearly, the t —>oo subspace lim ptE is canonically isomorphic to Em. t—too

If £0 is a continuous unit, but not necessarily unital, then we know from Reference3 how to normalize £0 to a continuous unital unit within E® . Moreover, if £0 is among the continuous sections, then so is its normalization. (The normalization suggested in Reference3 is unique, but even if there are several possibilities to obtain a unital unit in the subsystem generated by £ 0 , then the results below show that the continuous structure does not depend on the choice.) Henceforth, we assume that £® is unital. Denote by k = (&t) teK the family of canonical embeddings kt: Et —> £00- This provides us with a set CSk(EQ) of continuous sections as in Definition 7.1. 7.5 Theorem. Let £® be a continuous (unital) unit in a continuous product system E® . If £0 is in CSi(EQ), then the Bo-semigroup -d constructed on 33a(-Eoo) is strictly continuous and CSk(EQ) = CSi(E&). In particular, the continuous structure CSk(E®] of E® does not depend on £ 0 . Conversely, if £0 £ CSi(EQ), then, of course, CSk(EQ) PROOF. By the preceding discussion we may suppose that £0 is unital. As the case £0 <£ CSi(EQ) is clear, we suppose that £® e CSi(EQ). First, we show that i) is strictly continuous. The following proof is an imitation of that in Reference6 for type I systems, except that now we have to consider continuous sections which are not necessarily units. For a fixed y £ E^ the mapping t H-> srty := y 0 £t is continuous. (Indeed, as s£ is bounded by ||6|| = 1, it is sufficient to show the statement on the total subset of -Boo consisting of elements of the form y = ksys for some s 6 R + , ys 6 Es. By definition there exists x € CSi(EQ) such that ys = xs. Then (for e > 0; £ < 0 is analogous) \\ksya Q(,t+e-kayaQ>£.t\\

= \\ya ©6 ©6 - Ce Oy* 0611

< \\is+£(xs Q ^) - is(xs 0 £0)|| + ||^(^o 0 xs) - ie+a(£E 0 x (7.c)jf £; ancj E^ are isomorphic, but not necessarily canonical, then we know from Reference16 that the Eo-semigroups on E and £00 are cocycle conjugate.

239

is small for £ sufficiently small.) It follows that \\0t(a)y - ay\\ < \\tft(a)(y -y ©&)|| + \\ay 0& - oj/|| is small for t sufficiently small. This implies, in particular, that CSk(EQ] is the continuous structure derived from an ^Q-semigroup and, therefore, the product of sections in CSk(EQ) is continuous in the sense of Definition 7.1(2). Let x £ CSi(EQ). We find \\kt+EXt+E - ktxt\\ = \\Xt+€ - & Q Xt\\

= \\it+e(£o © Xt+e) - it+e(£e © Xt)

which is small for e sufficiently small so that x e CSk(EG). Conversely, let x € CSk(EQ). For fixed t 6 R+ we may choose y £ CSi(EQ) such that yt — Xt for that t. Let £ > 0. Left continuity we see from \\it+EXt+E ~ itXt\\

< \\it+E(Xt+E - £e O Xt)\\ + \\it+E(£E Q yt) - ityt\\

which is small for £ sufficiently small. For right continuity we observe that for small £ we have Xt « ^£ 0 xt_ e , because £e,x £ CSk(E®), and Vt w 6 ©y t _ e , because ^ 0 ,y e CSi(E®). Therefore,

and also \\it-eXt-e ~ itXt\\

< \\it-E(xt-e - yt-s)\\ + \\it-EVt-e ~

is small. • 7.6 Definition. A continuous product system EQ is type l(ll)(lll), if there is a generating continuous subset S C CSi(EQ) of units (if there is a continuous unit £0 € CSi(E&}} (if there is no continuous unit in CSi(EQ)). The following is a simple corollary of Theorem 7.5 and Proposition 7.4 7.7 Theorem. Non-type III continuous product systems are uniformly continuous. If E® is a non-type III product system, then, unless specified otherwise explicitly, we assume that it comes shipped with its natural continuous structure CSk(EQ). If EQ is continuous with respect to this structure, then type and continuous type coincide. (If specified differently, i.e. if the continuous unit making the product system non-type III, the types need not coincide.)

240

7.8 Example. The product system in Example 4.7 is continuous type III, although its unit is among the continuous sections. It is an interesting problem to find examples for a continuous product system of Hilbert modules without units in CSi(EQ). So far, we have to content ourselves with Tsirelson's Hilbert space example. Apparently, for Hilbert modules there are different levels of type III. Definition 7.1 excludes some interesting product systems of von Neumann modules (or even of Hilbert modules, when B is a von Neumann algebra or otherwise too big). For instance, the product system of the time shift s on the strong closure B = L°°(R+) of Example 4.7 has not a single non-zero continuous section, (s is only normal, but not Co-continuous.) Of course, a product system arising from such a natural semigroup like the time shift should belong to the objects of interest, so we have to find a definition which suits better for von Neumann modules. A first possibility is to replace everywhere in Definition 7.1 'continuous' with 'strongly continuous'. A second possibility is based on the following observation. 7.9 Proposition. Let EQ be a product system of Hilbert B-B-modules with a family i = (*t) teR of embeddings it: Et —» E into a united Hilbert Bmodule and define CSi(E&) as in Definition 7.1. Suppose CSi(EQ) fulfills Condition (2) and the weaker condition 1'. For every t e R + the subspace CSt = {xt (x £ CSi(E®))} is dense in Et. Then E® is a continuous product system. PROOF. Fix t S K + . We have to show that the subspace CSt is all of Et. We recall a well-known result from Banach space theory. If W is a dense subspace of a Banach space V, then for every v £ V there exists a 00

sequence (wn}n€N in W such that the series ^ wn converges absolutely 71=1

to v. So, for xt € Et choose a sequence (s ra ) ngN in CSi(E&) such that the 00

series ^ x™ converges absolutely to x t . We may assume (possibly after n=l

multiplying each section xn by a suitable continuous numerical function) that ||o;" || < ||o;"|| for all s, n. It follows that the series over the sections xn converges uniformly over s € R+ to a continuous section x with the correct value xt. •

241

If we replace in Definition 7.1 Condition (1) by Condition (!'), then we obtain a definition suitable for von Neumann modules, if we replace further in Condition (!') 'dense' by 'strongly dense'. In both possible definitions our example L°°(1R) would be continuous. Here we do not intend to decide between the possible definitions. A decission should be based on further investigation of examples and, in particular, of counter examples. We close with a remark on continuity of units in product systems of von Neumann modules. By a recent result of Elliott8 Co-convergence to \dB of a sequence of normal completely positive mappings on a von Neumann algebra implies uniform convergence. It is routine extension of this result to conclude that every normal Co-continuous CP-semigroup is uniformly continuous. Therefore, in a continuous (in the sense of Definition 7.1) product system of von Neumann modules the set U(EQ) D CSi(E&) is a continuous set of units. Acknowledgements This work is supported by a PPP-project by DAAD and DST. References 1. W. Arveson. Continuous analogues of Fock space. Number 409 in Memoires of the American Mathematical Society. American Mathematical Society, 1989. 2. W. Arveson. Continuous analogues of Fock space III: Singular states. J. Operator Theory, 22:165-205, 1989. 3. S.D. Barreto, B.V.R. Bhat, V. Liebscher, and M. Skeide. Type I product systems of Hilbert modules. Preprint, Cottbus, 2000. 4. B.V.R. Bhat. An index theory for quantum dynamical semigroups. Trans. Amer. Math. Soc., 348:561-583, 1996. 5. B.V.R. Bhat and K.R. Parthasarathy. Kolmogorov's existence theorem for Markov processes in C*-algebras. Proc. Indian Acad. Sci. (Math. Sci.), 104:253-262, 1994. 6. B.V.R. Bhat and M. Skeide. Tensor product systems of Hilbert modules and dilations of completely positive semigroups. Infinite Dimensional Analysis, Quantum Probability & Related Topics, 3:519-575, 2000. 7. E. Christensen and D.E. Evans. Cohomology of operator algebras and quantum dynamical semigroups. J. London Math. Soc., 20:358-368, 1979. 8. G.A. Elliott. On the convergence of a sequence of completely positive maps to the identity. J. Austral. Math. Soc. Ser. A, 68:340-348, 2000. 9. G.G. Kasparov. Hilbert C**-modules, theorems of Stinespring &; Voiculescu. J. Operator Theory, 4:133-150, 1980. 10. B. Kiimmerer. Markov dilations on W*-algebras. J. Fund. Anal., 63:139177, 1985.

242

11. E.G. Lance. Hilbert C*-modules. Cambridge University Press, 1995. 12. V. Liebscher. Random sets and invariants for (type II) continuous tensor product systems of Hilbert spaces. Preprint, 2001. 13. V. Liebscher and M. Skeide. Units for the time ordered Fock module. Infinite Dimensional Analysis, Quantum Probability & Related Topics, 4:545—551, 2001. 14. Y.G. Lu and S. Ruggeri. A new example of interacting free Fock space. Preprint, Rome, 1998. 15. N. Muraki. Monotonic independence, monotonic central limit theorem and monotonic law of small numbers. Preprint, Takizawa, 2000. To appear in Infinite Dimensional Analysis, Quantum Probability & Related Topics. 16. M. Skeide. Dilations, product systems and weak dilations. Preprint, Cottbus, 2000. To appear in Math. Notes. 17. M. Skeide. Generalized matrix C"*-algebras and representations of Hilbert modules. Mathematical Proceedings of the Royal Irish Academy, 100A: 11-38, 2000. 18. M. Skeide. Hilbert modules and applications in quantum probability. Habilitationsschrift, Cottbus, 2001. Available at http://www.math.tu-cottbus.de/INSTITUT/lswas/_skeide.html. 19. M. Skeide. The index of white noises and their product systems. Preprint, Rome, 2001. 20. B. Tsirelson. From random sets to continuous tensor products: answers to three questions of W. Arveson. Preprint, ArXiv:math.FA/0001070, 2000. 21. B. Tsirelson. From slightly coloured noises to unitless product systems. Preprint, ArXiv:math.FA/0006165, 2000.

QUASI-FREE FERMION PLANAR QUANTUM STOCHASTIC INTEGRALS

W. J. SPRING Department of Computing and Information Systems, Faculty of Science, Technology and Design, University of Luton, Park Square, Luton, L Ul 3JU, UK E-mail: [email protected]

I. P. WILDE Department of Mathematics, King's College London, Strand, London, WC2R 2LS, UK E-mail: [email protected]

Quantum analogues of planar stochastic integrals of the first and second kind, as introduced by Wong and Zakai, are constructed for a quasi-free theory of fermions.

1. Introduction We consider a quantum stochastic calculus in which the parameter space for the nitration is a square, which we may as well take to be the unit square [0,1] x [0,1] in the positive quadrant of the plane. Filtrations of subalgebras of an operator algebra or subspaces of a Hilbert space replace the usual filtration of cr-algebras, as in the classical stochastic theory. The integrators of our theory are given by quasi-free fermion creation and annihilation processes. Using these, we construct various fermion quantum analogues of the planar stochastic integrals of the first and second kind as constructed by Wong and Zakai.1 A quantum analogue of such integrals has already been constructed in terms of a two-parameter Ito-Clifford type theory2 by Spring and Wilde3. Stochastic integration in the plane has been studied by Wong and Zakai1 and further developed by Cairoli and Walsh4. A recent discussion of classical planar stochastic integration has been given by Green5.

243

244

2. Quasi-free aspects of the CAR Let Q denote the unit square [0, 1] x [0, 1] in the plane R 2 and let #o denote the antisymmetric (fermion) Fock space over f) = L2(Q) (with Lebesgue measure). Set H = $0 $0 and, for / . g e t ) define the fermion creation and annihilation operators 6* (/) and b(g) on H by the formulae 1/2 1/2

&•(/) = &S((i - p) f) ® i + r(-i) ® to(p 7)

(i)

and b(g) = 6o((l - P)1/2S) ® 1 + F(-l) ® &S(/>V2S)

(2)

where &Q and 60 are the Fock space fermion creation and annihilation operators, respectively, the function p obeys 0 < p(w) < 1 on Q, and F(— 1) is the usual Fock space parity operator. (F(— 1) is defined on the fermion Fock no-particle vector QO by F(— l)£io = ^o and is given on each n-particle subspace of Jo by <£>"(—!).) The operators 6* and b obey the canonical anticommutation relations (CAR)

b(f)b(g) + b(g)b(f)

= &*(/)&*($) + b*(g)b*(f)

=0

for any f,g£ [). Notice that b*( • ) is linear in its argument, whereas b( • ) is conjugate-linear. Let 21 be the unital C*-algebra generated by the 6*(/)s and b(g)s as / and g vary over f). Set fi = fi0 <8> ^o and let u> be the vector state w( • ) = ( • fi, fi) on 21. Then w defines a gauge-invariant quasifree state on the CAR C*-algebra. u is such that w(6*(/)) = w(b(g)} = 0 (gauge-invariance), and the two-point functions are given by (pf,g)t)

(4)

for all /,# G f). Furthermore, the value of w(6*(/i) . . . b*(fn)b(gi) . . . b(gm)) is zero if n ^ m and otherwise is given in terms of various products of the two-point functions u>(b*(fi)b(gj)). There are two consequences of this that we wish to note here. The first is that if, say, fk has support which is disjoint from (or has overlap of measure zero with) each of the supports of the functions g^ (or, similarly, if gr has no overlap with any of the /js), then w(6*(/i)...6*(/ n )fe(ffi)...6(Sm))=0.

(5)

This is simply because all the two-point functions uj(b*(fk)b(gj)), 1 < j < m (respectively, u(b*(fi)b(gr)), 1 < i < n) vanish. The second property that we wish to highlight is the following product structure of w. Suppose that

245

/ € f) has support whose overlaps with the supports of each of /j , , , . , /n and
(6)

where 6# stands for either b or b*. For further details and references on quasi-free representations of the CAR, we refer to the article of Powers and St0rmer6 as well as, for example, the account given by Bratteli and Robinson.7 3. Quasi-free filtration We use the word rectangle to imply that its sides are parallel to the coordinate axes. For any z = (zi, z 2 ) £ Q, let us denote by Qz the rectangle [0, zi] x [0, 23] in <5, that is, Qz has bottom left-hand corner at the origin and top right-hand corner at the point z. The square Q is given the partial order z < w defined by the rule z<wifQz C Qw. This simply means that z < w if Zi < Wi, for i = l,2, where z = (21,2:2) and w = (wi,w2). For any rectangle R in the square Q, inf R is the bottom left hand corner of R. For z € Q, let 21^ denote the C*-algebra generated by the 6*(/)s and b(f)s as / runs over those elements of t) with support in the rectangle Qz. Clearly, 212 C 21^ whenever z < w so that { 2lz : z 6 Q } is a filtration of C*-algebras. The CAR algebra, 21, is generated by the family { 212 : z e Q }. Denote by Az the von Neumann algebra 21" generated by 2lz and let A = 21". Let "Hz be the (closed) subspace of "H generated by the linear space 21ZQ (and so Hz is also generated by AZSI). Then { Az : z £ Q } and { T~tz '. z £ Q} are filtrations of von Neumann algebras and Hilbert spaces, respectively. For each z e Q, there is an w-invariant conditional expectation from 21 onto 2lz which extends to an w-invariant normal conditional expectation E z : A —> Az. Moreover, if Pz : H —> Hz denotes the orthogonal projection, then Pzx£l = Ez(x)fl for any x € A. The vector fi is cyclic and separating for A (and therefore also for 21). For further discussion of these matters, we refer to the articles of Evans8, Takesaki9 and Barnett, Streater and Wilde10. If A is a subset of Q, then for notational simplicity we shall use A to stand for both the set A itself and also the corresponding indicator function of the set A. Let bf = b#(Qz), where we recall that 6* stand for either 6* or b. Then Ez(b#(Qw)) = b#(QzQw) so that the families { bf : z € Q } are

246

21-valued martingales with respect to the ordering < on the two-parameter space Q. As shown by Wong and Zakai,1 there is another partial order on the unit square, Q, which is pertinent to the theory. Let us say that z A w if z\ < w\ and z^ > u>2, where z = (zj, 2:2) and w = (wijW^). In other words, z A. w means that the point z is above but to the left of the point w. For rectangles 5 and R, we write S A R to indicate that z A w for each z £ S and w e R. Given rectangles S A R, let us put /u(5, R) = inf 5vinf .R, that is, n(S,R) is the maximal point in Q (with respect to the partial order < ) such that QH(S,R) is has overlap of measure zero with each of S and R. Explicitly, if inf 5 = (zi,Z2) and inf R = (wi,wz), so that z\ < w\ and z-i > u>2, then n(S, R) = (wi, z2). 4. Quantum stochastic integrals In this section, we construct fermion quantum analogues of Wong and Zakai 's stochastic integrals of the first and second kind using the 21-valued martingales b* and bz as integrators. We begin with some definitions. Definition 4.1. A map 77 : Q —> H is said to be an elementary adapted W-valued process if it has the form (recall that XR(Z) is just written as R(z)) •q(z) = R(z}^,

zeQ,

(7)

for some rectangle R C Q and for some i/J € T~imfR- In the obvious way, one defines 21-valued or A-valued such processes (by simply demanding that ip belongs to 2t; n f/j or to AM R, respectively). A finite linear combination of elementary adapted processes is called a simple process. Note that if 77 is, say, a simple ,4-valued process, then 77^ is a simple W-valued process. Definition 4.2. For z € Q and elementary adapted process rj(w) — R(w)il;, the quantum stochastic integrals of the first kind, JQ db# rj(w), are defined

by .

(8)

The definition of these quantum stochastic integrals is extended by linearity to the collection of simple processes. Notice that if 77 is 2t-valued, or ,4-valued or H-valued, then so are the stochastic integrals, respectively. This definition is a quantum version of the

247

classical one-parameter Ito-integral. The term b#(RQz) is the "integrator increment" which "points to the future" . The observation of Wong and Zakai1 is that in a two-parameter theory one can construct a new type of stochastic integral. For this, one uses double integrals based on the partial order A. on Q, introduced above. Definition 4.3. The map rj : Q x Q —> H is said to be an elementary adapted process if it has the form 77(5, t) = S(s)R(t)ip for rectangles S A. R and ip £ T~tp.(s,R)- For z G Q, we define the four quasi-free fermion quantum stochastic integrals of the second kind by

Q, Q*

db#db?r)(s,t)=b#(QzS)b#(Q,R)1>.

(9)

As before, this construction is extended by linearity to simple processes r](s,t). We could also consider 21- valued or A- valued processes, as above, but we will only discuss the 7^-valued case here. It turns out that all these quantum stochastic integrals are martingales. Theorem 4.4 (Martingale Property). Let h(s) be a simple process. Then, for any w < z, Ew(f

^JQ*

dbfh(s)} = I

'

JQW

dbth(s),

(10)

if h is a Vi-valued or A-valued adapted process, or dbfh(s)) '

= I dbfh(s), JQv,

(11)

if h is an H-valued adapted process. Furthermore, for any simple process

Pw f

I

•/Q* JQ*

dbfdbfr](s,t)=[

I

dbf dbf r,(s,t).

(12)

JQu, JQW

Proof. Let w < z. By linearity, it is enough to consider elementary adapted processes. Consider, first, the case where h is Ti-valued. Then h is of the form h(s) = R(s)ip for some rectangle R C Q and i/j £ H[nfR. Now, if Qw n R has measure zero, then /„ dbf h(s) = b#(QwR)i{> = 0. Otherwise, we have inf R < w and so /„ dbf h(s) = b#(QwR)i() belongs to HwTherefore, it is enough to show that (fQ d b f h ( s ) , £ ) = (fQ dbfh(s),£)

248

for any f e Hw. However,

/ db* h(s) - I dbf h(s) = b*(QzR}^ - b*(QwR)^ JQ, JQ™

(13)

and so we need only verify that (b#((Qz\Qw)R)tl>, £) = 0. Now, by linearity and continuity, we may suppose that if) is given by V = b#(gi) . . . b#(gn)tl where the test-functions QJ 6 [) have support in the rectangle Qw. (The value n = 0 is allowed, in which case i/j — fl.) The conclusion, namely eqn(ll), now follows directly from the lack of overlap, eqn(5). For the case when h is 21-valued or .A-valued, we simply note that h£l is W-valued and so, from the part above,

Ew( [ dbf h(s)}tl = Pw [ dbfh(s)£l=[

VQ,

'

JQ*

JQW

dbfh(s)Sl

(14)

and we deduce that E^ f /„ dbf h(s) ) = JQ dbf h(s) since fi is a separating vector for A. This establishes eqn(10). To demonstrate the martingale property for the integrals of the second kind, eqn(12), we note that, by linearity, it is enough to consider r\ of the form rj(s, t] = S(s)R(t)ip where 5 and R are rectangles in Q with S A R and if} € T~i/j.(stR). By linearity and continuity, we may also suppose that V> = x£l where x = b ^ ( f i ) .. . 6*(/m) for some m > 0 and for fj € f) with support in Q^(S,R) for all j. (The case m = 0 corresponds to x = 1.) Let ip = y£l, where y has the form y = b&(g\) . . . b&(gn) for gi in f) with support in Qw. (As above, y = 1 when n = 0.) Such ips are total in "Hw and we need only show that (f f dbfdbtr,(s,t),v)=([ f dbfdbfr,(s,t),
[ dbfdbfr,(s,t),V)-(f

JQ, JQ*

f

JQW JQW

= ((b#(Q2S)b#(QzR) -

(15)

dbfdbfr,(s,t),V)

b*(QwS}b#(QwR)W,V)

- (^((Q^Q^S^Q^x^ySl) + (b*(QwS)b#((Qz \ Qw)R)xSl, yfl)

(16)

+ (b*((Qz \ QwS}b*((Qz \ Qw)R))xSl, yfl) = 0,

by eqn(5), as required.

249

If h(s) = Q(s)fi for s € Q, then we have /Q^ d&f /i(s) = 6#(<5 z )fi and so || f jQz

If b* = b, eqn(17) gives || /Q^6S /i(s)|| 2 = fQfP(w)dw. On the other hand, if 6# = b*, we use the CAR to write b(Qz)b*(Qz] = fQ Qz(w)dw b*(Qz)b(Qz) so that eqn(17) then gives the equality || fQzdb*s h(s)\\2 = JQ (1 — p(w)) dw. These formulae generalize to give isometry relations obeyed by the various quantum stochastic integrals. Let p# = p when 6* stands for 6, otherwise let p& = (1 — p) (when 6* =6*). Theorem 4.5 (Isometry Property). Let h(s) be any given simple process. Then, if h is A-valued, u(( [ db*h(w)Y f db*h(w)}= f w(h(w)*h(w))p#(w)dw, ^JQ, ' JQ, JQ,

(18)

whereas if h is H-valued, then )\f=

Q*

JQ,

\\h(w)\\2p#(w)dw.

(19)

Also, for any simple process 77 : Q x Q —> H, \ \ f f dbfdbt r,(v,w) "JQ,JQ,

2

= / f \\r,(v,w)\\2p#(v)p#(w)dvdw. JQ,JQ*

(20)

Proof. We begin with eqn(19) and suppose h to be W-valued. The Avalued situation, eqn(18), reduces to this case. By further refinement, if necessary, we can assume that h may be expressed as h = Y^™=i Wj wi*n Wi — fijVj and where the rectangles RI,.. ., Rm are disjoint except, possibly, for intersecting edges. By linearity and continuity, we may also assume that •03 = Xj£l where Xj = b#(gji) . . .b#(gjm.} for suitable elements gji e f) with support in QmtRj- The left hand side of eqn(19) is a sum of inner products of the form (/^ db# rjj(w), fQ^ db*, rjk(w')). Suppose j = k and suppose, first, that the integrator 6* stands for bw. Then the inner product

250

above becomes

Q,

dbwTlj(w),

JQ, = 1 \Qz(w)Rj(w)\2 p(w)dwu(xjXj), JQ = I

using eqn(6),

Rj(w)\\^jfp(w)dw.

(21)

JQ,

If b* = b*w, we use the CAR to write (b*(QzRj)xjV:, V(QzRj)xj^l)

as

= f \Qz(w)Rj(w)\* dwu JQ

Q

\Qz(w)Rj(w)\2 (1 -p(

= f

Rj(W) \\fa\\2 (I -P(w))dw.

(22)

^Q,

Now suppose that k ^ j. Then the rectangles Rj and Rk are disjoint (or possibly have overlapping sides) and therefore their interiors lie either side of either a horizontal line or a vertical line (which may pass through a side of a rectangle). It follows that at least one of Rj or Rk has overlap of measure zero with each of Q-iniRi and Q-mfRk- Hence ( f dbt^w), f JQ, JQ* =0

(23)

by eqn(5). Finally, using eqn(21) or eqn(22) and eqn(23), we get

Q,

jtk

, f db#r,k(w')) JQ, 2

dw

(24)

= / P#(w)\\r,(w)\\*dw, JQ, as required. To establish eqn(20), let 77 = 2"=1r/j with r)i(v,w) = Si(v)Ri(w}fa where 5, A Ri and fa e "H^s^Ri)- By linearity and continuity, we may

251

assume that V'z = Xi$l, where Xi is either equal to c^l, for some a.i € C, or else is a product of b#(g)s, with the supports of the gs contained in Qv(Si,Ri)- Furthermore, we may suppose that the rectangles are such that, for any i ^ j, either (i) Sj D Sj and Ri n Rj both have measure zero, or (ii) Si = Sj and Ri n Rj has measure zero, or (iii) Ri = Rj and Si n 5, has measure zero. Then it is certainly the case that that at least one of Si n Sj or Ri n .Rj has measure zero. The left hand side of eqn(20) is equal to I dbfdb* r,i(v,w), I

,JQ*

I

JQ*JQ,

dbtdb#r,j(v,w)).

(25)

We will show below that the off-diagonal terms (i ^ j) all vanish. The typical diagonal term is equal to

= =

u(x*b#*(QzRi)b#*(QzSi)b#(QzSi)b#(QzRi)xi) Jj'(QzSi)(v}(QzRi)(w)p#(v)p#(w}ij(x*Xi)dvdw,

using eqn(6) (and, possibly, the CAR),

Summing over i gives the isometry property, eqn(20). To complete the proof, we must show that, indeed, the off-diagonal terms vanish. Such a term is (with i ^ j)

= u(x'ib#*(QtRj)b#'(QxSj)b#(QxSi)b#(QtRi)xi').

(26)

Now, suppose that Si and 52 are essentially disjoint, i.e., their intersection has measure zero. This implies that there is some horizontal line or some vertical line (or both) separating their interiors. Suppose the former, and let us further suppose that Si lies above £2. Then Si is positioned above RI and also above R
252

Proof. First, we will show that the quantum stochastic integrals SQ, dbw h(w) and fQ t db*w h'(w) are orthogonal. We may suppose (following further refinement, if necessary) that h and h' have the form h(w) — T,T=i Ri(w)i>i and h'(w) = £™ 1 Ri(*>Wi for rectangles Ri,...,Rm such that Ri n RJ has measure zero for all i ^ j. But then, by linearity, it is enough to consider the case h(w) = R(w)if} and h'(w) = R'(w)tp' where either R = R' or else R and -R' have overlap of measure zero. We may also assume that both if) and tfj' have the form x£l and x'fl, respectively, where x and x' are products of various b#(g)s with the gs having support in Q\nfR or Qinffi') respectively. In this case ( / dbw h(w), [ db*w ti(w)) = (b(QzR)xfl, b*(Qz,R')x'Sl). (27) JQ, JQ,, Now, if R = R', the right hand side of eqn(27) vanishes, by eqn(5). On the other hand, if R and R' have overlap of measure zero, then, as noted earlier, they can be separated (up to a set of measure zero) by either a vertical or a horizontal line. Then certainly Q2/ n R' and Qz n R have overlap of measure zero and at least one of them also has overlap of measure zero with each of Qinf R and Qinf R'. By eqn(5) once again, it follows that the right hand side of eqn(27) is zero. To show the orthogonality of /„ db# h(w) and /„ /;JQ (/ dbfdb# TJ(V, w), it is enough to consider elementary adapted integrands. To this end, let h(w) = S'(w)tp' and TI(V,W) = S(v)R(w)i{> be elementary adapted processes. We may further assume that tp' = x'fi and that ip = y£l where x' and y are products of 6#(g)s where the gs have support in Qj n f s> or Q^(s,R)-> respectively. We may also assume (possibly following further refinement) that S' = S or S' = R or else S' is disjoint from both S and R (up to sets of measure zero). The inner product of the two stochastic integrals is then

= u(y*b#*(Qz,,R)b#*(Qz»S)b#(Qz,S')x')

(28)

which vanishes, by eqn(5), in all of these possible cases. Similarly, suitable application of eqn(5) is used to show the orthogonality of the other pairs of stochastic integrals. Let K,# denote the completion of the set of simple Ti.- valued processes with respect to the Hilbert space norm \\h\\ = (JQ ||/i(iu)||2 p* (w) dw) . The isometry property, eqn(19), allows the definition, by continuity, of the quantum stochastic integrals /^ db#f(w), for any / € K,#. Theorems 4.4, 4.5 and 4.6 each extend to this situation, as summarized in the following.

253

Theorem 4.7. For any f € K. and g £ K,*, the fermion quantum stochastic integrals JQ dbwf(w) and / Qj db*wg(w) are H-valued martingales which satisfy the isometry properties \\f

JQ*

db#f(w)\\2 = f

\\f(w)\\* p#(w)dw.

(29)

JQ,

Furthermore, f ^ dbwf(w) is orthogonal to JQ

t

db*wg(w).

Remark 4.8. Similarly, one can extend the definition of the quantum stochastic integrals of the second kind to integrands in appropriate completions. Once again, these are also martingales obeying isometry relations and pairwise orthogonality. Remark 4.9. There is a corresponding construction of planar quantum stochastic integrals of the first and second kind within a quasi-free theory of the CCR. In this case, the integrators are boson quasi-free creation and annihilation processes. The details will appear elsewhere.

References 1. E. Wong and M. Zakai, Z. Wahrscheinlichkeitstheorie verw. Gebiete 29, 109 (1974). 2. C. Barnett, R. F. Streater and I. F. Wilde, J. Functional Analysis 48, 172 (1982). 3. W. J. Spring and I. F. Wilde, Reports on Mathematical Physics 42, 389 (1998). 4. R. Cairoli and J. B. Walsh, Acta Mathematica 134, 111 (1975). 5. M. L. Green, Planar Stochastic Integration Relative to Quasimartingales in Real and Stochastic Analysis, Recent Advances, Ed. M. M. Rao, CRC Press, New York, 1997. 6. R. Powers and E. St0rmer, Commun. Math. Physics 16, 1 (1970). 7. O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics II (Springer- Verlag, New York, 1981). 8. D. E. Evans, Commun. Math. Physics 70, 53 (1979) 9. M. Takesaki, J. Functional Analysis 9, 306 (1972) 10. C. Barnett, R. F. Streater and I. F. Wilde, J. Functional Analysis 52, 19 (1983).

ANTILINEARITY IN BIPARTITE QUANTUM SYSTEMS AND IMPERFECT QUANTUM TELEPORTATION

ARMIN UHLMANN Institute for Theoretical Physics, University of Leipzig E-mail armin. uhlmann@itp. uni-leipzig. de

Antilinearity is quite natural in bipartite quantum systems: There is a one-to-one correspondence between vectors and certain antilinear maps, here called EPRmaps. Some of their properties and uses, including the factorization of quantum teleportation maps, is explained. There is an elementary link to twisted Kronecker products and to the modular objects of Tomita and Takesaki.

Introduction In this paper I consider some assorted antilinear operations and operators in bipartite quantum systems, an application to quantum teleportation, and a link to Tomita and Takasaki's theory via twisted direct products. The idea is in exploring the natural antilinearity which is inherent to vectors in direct products of Hilbert spaces. The reason for the appearance of certain antilinear maps, here called EPR-maps, is explained in the first section, together with some basic equations. The acronym EPR stands for the problem, raised by Einstein, Podolski, and Rosen1, see also Peres3, Nielsen and Chuang.4 Antlinearity in the EPR-problem has been explicitly noticed by Fivel.5 Here I follow a more general line.6 7 Of course, the exposition in the first section (and in the third one) are mathematically near to almost every treatment in which purification and related topics play there role. Antilinearity is often masked by introducing distinguished basis in the parts of the bipartite system. An interesting different approach is by Ohya and Belavkin,8 9 and by Ohya's idea10 of compound states. In section 2 I present an application to imperfect (unfaithful) quantum teleportation: Linear teleportation maps allow for a unique decomposition into pairs of EPR-maps. Uniqueness would be lost by requiring linearity due to an ambiguity in phases. 255

256

Two norm estimates are derived. The case of Liiders measurements with projections of any rank is considered. An example with distributed measurements is presented, showing the use of antilinear EPR-maps in a multipartite system. The polar decompositions of EPR-maps are considered in section 3, a rather straightforward task. In these decompositions the positive parts are the square roots of the density operators seen in the two subsystems. The phase operators must be antilinear partial isometries between the two parts of the direct Hilbert space product. As explained in section 4, this feature allows to perform twisted direct products. They will be compared with an elementary case of well known operators known from Tomita-Takesaki theory. In view of applications to quantum information theory, and to underline the difference to classical intuition, one often assumes a macroscopic distance between the two systems. Though this is reflected in the formalism only rudimentarily, it provides a nice heuristics: The subsystems can be distinguished classically, their owners, Alice and Bob, can exchange classical information (using, say a telephon), and they are independent one from another. If they like to perform quantum operations, they have access just to their parts. Notice that a macroscopic spatial distance between them is sufficient for the observables of Alice to belong to the commutant of Bob's observables. Of course, parts of a composed quantum system can be independent one from another without sitting in spatially different regions. Remarks on notation: In this paper the Hermitian adjoint of a map or of an operator A is denoted by A*. The scalar product in Hilbert spaces is assumed linear in its second argument. Sometimes the symbol o is used to see more clearly how maps are composed. 1. Some basic facts Our bipartite quantum systems lives on the direct product H := Ha®Hb of two Hilbert spaces, T~ia and Hb, with any dimensions. (A nice little exercise is to follow the formalism in case of a 1-dimensional Hb-) It is a well known fact that H is canonically isomorphic to the space of Hilbert-Schmidt maps from Ha into the dual HI of Hb-

H = Ha®Hb^ C2(Ha,HI) ^ £2(Hb,K) HI is antilinearly (or conjugate linearly) isomorphic to Hb, a fact which is on the heart of Dirac's bra- ket-formalism \x) <-> (x\. Composing the bra-ket morphism with the Hilbert-Schmidt maps from Ha into HI we get

257

the space of antilinear Hilbert- Schmidt maps from Tia into Tib- Indicating the antilinearity by an index anti, we have the natural isomorphisms

Let us look at these morphisms in more detail, and let us start with an arbitrary vector i/> from H. There are decompositions

fc®0t # € « „ , $ € %

(2)

converging in norm. Choosing one of them arbitrarily, we set s>°:=I> 0 ,#>$

(3)

Every member of the sum is a map from Tia into *H\,. Their 2-norms are the same as the norm of the corresponding term in the decomposition (2). Hence, (3) defines an antilinear Hilbert-Schmidt map from Ha into Hb. Its adjoint, a map from Tib into Ha, is defined by the relation (/,s^0} = (r,(s£T/) a

(4)

b

for all and 4> . By an evident calculation one gets

and we denote this map in accordance with (3) by s°£. In the next step we explicitly see the independence of the constructions from the chosen decomposition (2) of ty. It provides the contact to a famous problem of Einstein, Rosen, and Podolski.1 Assume the state of the bipartite system is defined by ip & ~H. If Alice does a measurement with one of her observables, A € B(Ha), her activity is a measurement in every larger quantum system which contains Alice's system. In particular, this is the case in the bipartite system based on Ji. Here the relevant observable reads A® I6. We now choose Alice's observable to be the rank one projection P = l^ >a ){ < /' a |) <^a € ^a being a unit vector. In doing so, the measurement terminates in showing randomly the eigenvalue 1 or 0 of P. In case it shows the eigenvalue 1, the state vector of the bipartite system system has switched from -0 to (P ® I6)'!/'- A new state vector has been prepared. Our aim, to show the independence of (3) from the chosen decomposition (2) of if}, is reached by proving

(| a ' V^ a eW a

(6)

To show (6) for a given decomposition of ifr, one first remarks the linear dependence of (3) from the terms of the sum (2). Thus, one has to check

258 (6) just for product vectors, a simple task. Remark that a similar relation holds for an appropriate action of Bob. In conclusion we have seen that every if> £ 7i uniquely determines antilinear Hilbert-Schmidt maps according to (3) and (5). Let us call them the EPR-maps belonging to if>. They are antilinear equivalents of ip obeying ba ab (c S \* — Sa

(, V ) — V '

(aab\* — «bo

(,SV> ) — SV>

I7\

(')

In rewriting (4) and (7), we can add a conclusion seen from (6): It holds {/, s$") = (a ) a

(8)

b

for all (f> 6 Ha, (f> G Hb, and ip € H. Now we proceed as follows: Because every tp € H can be written as a sum of product vectors, we try to calculate its scalar product with if) by the help of (8). A more or less straightforward calculation will show the validity of

(v,i>)=TraSfSbva=TrbS$s«b

(9)

the right hand term of which are, in view of (7), antilinear versions of the von Neumann scalar product. Let me add that one can derive (8) from (9) by choosing (p = (j>a ® 4>b. What remains for a first account is the reconstruction of ip from one of its EPR-maps. The task can be done with the help of any decomposition of the unit operator la of, say, Alice. More generally, let A € B(H.a) be a positive operator and

a rank one decomposition of A. Then

^ = £>2®s>2

(ii) a

if) is returned with Alice's unit operator, A = l . The reduced density operator, w^,, can be defined by

Tra A^ = ^,(A®lbW),

Ac B(Ha)

Similar one gets u>^ by letting play Bob the role of Alice. What one can learn from (6) and (7) is . ,o _ _a6 ba ^V ~ Si/> -0 '

6 _ i/> ~

6a ab -0 ip

/-r n\ \ >

Finally we consider two vectors which are related by y> = (A 5)V>

(13)

In terms of EPR-maps the relation converts to s^Bs^M*,

s$ = AB$B*

(14)

259

2. Imperfect quantum teleportation Bennett et al.n invented a protocol, the BBCJPW-protocol, allowing for faithful teleportation of vectors and of general states between Hilbert spaces of finite and equal dimensions d. It consists of one classical information channel and d2 quantum channels. The latter are randomly triggered by a Bell-like von Neumann measurement. The information, which quantum channel has been activated, is carried by the classical channel. It serves to reconstruct, by a unitary move, the desired state at the destination. The protocol has been programmed as a quantum circuit by Brassard.12 A general and self-consistent discussion of all perfect teleportation schemes and their relation to dense coding has been given recently by Werner.13 All these tasks and protocols need reference frames (computational basis) in order to define whether the original and the teleported vectors (or general states) should be considered as equal ones or not. Notice: the problem is not to tell which of the quantum channels is triggered nor to identify its output. It is the question how to relate the input to the output. Usually the problem is solved by distinguished reference basis, one in the input and one in the output space. Every reference base determines a conjugation. These conjugations, composed with the canonical antilinear maps, mask the natural antilinearity in all these protocols. Now I am going to describe the way antilinearity enters in the handling of general, possibly imperfect, unfaithful teleportation channels. Let Ji be a tripartite Hilbert space Uabc =Ha®Hb®Hc

(15)

The input is an unknown vector (f>a 6 Ha. One further needs a resource which provides the so-called entanglement2 between the b- and the c-system. The resource is given by an ancilla, mathematically just a known vector ipbc, chosen from Jii,®Jic. (More involved, but also tractable, is the case of an ancilla in a mixed state.) Thus, the teleportation protocol starts with a vector

vabc •= 4>a ®vbc e Habc

(16)

It is triggered by a measurement within the afr-system. We need a measurement which is also preparing. There should exist an apparatus doing it. But a single apparatus can only distinguish between finitely many values. The conclusion is: We have to trigger the protocol by measuring an

260 observable,

in the a6-system which is a finite sum with mutually different values a,j . The PJ are projection operators, orthogonal one to another, and decomposing the unit operator of T-iab- The measurement itself selects randomly one of these projectors with a well denned probability. If this projection is PJ, then the measuring device points onto the value a,, thus indicating which projection is preparing the new state. The duty of the classical channel is to inform the owner of the c-system which projection has been processing. For the discussion of the preparing we assume that P = Pab is one of the projectors PJ appearing in (17). A measurement in the a6-subsystem is simultaneously a measurement in the larger afec-system, and there the projection operator reads P® 1°. Thus, the preparing becomes a ® a ® vbc]

(18)

We now impose a restrictive assumption in (18): P should be of rank one. Thus, P has to test whether the afr-system is in a certain vector state, say if) = if)ab, or not. As the main merit of the assumption, the prepared state gets the special form

( r6}^06 ® ic)OT ® v>6c) = ^ab ® c,

(19)

determining 4>c € "He- Varying 4>a we now define the map t^° by

t^>a = c

(20)

The teleportation map t™ , or tca for short, can be computed6 by j-ca

_

cb

ba

*-il>,(p — s

fry-i \

(zl->

This is the factorization property, valid for every (imperfect) teleportation channel under the condition that the preparing projection operator is of rank one. There is no restriction otherwise, neither on the dimensions of the Hilbert spaces, nor on the ancillary vector

*, ^> • • •> °f ^fe we write, according to (11)

261

Next, this expression inserted into (19) yields

(8) allows to rewrite the scalar product and to get ^ ® ^ = £>$?
The antilinearity of the EPR-map converts the right hand side into

which is the assertion. Before looking at some applications of the factorization theorem, I mention that Alberio and Fei14 derived a condition for a generally imperfect channel to become faithful. 2.1. Estimates The high symmetry provided by maximally entangled vector states used in faithful teleportation schemes11 13 is broken in imperfect teleportation. As a result, some of the vectors in Ha are more efficiently transported than others. Therefore, the highest possible transport probability is of some interest. Let (j)a, if}, tf> be unit vectors. The probability for the process a —» (/>c is

Because ip and tp are vectors of two bipartite systems, and Ht, is a part of both systems, we may compare their reductions to the b-system. One can prove, see (40) and (41) below, (^,0 C )<|(^) 1 /^ ( w 6 ) l/2 | o o

(22)

c

for all unit vectors in 6 Hc- The norm used at the right hand side is the operator norm. The norm of a positive operator is its largest eigenvalue. Being of trace class, one would like to estimate the effectivity of the single teleportation map by the trace norm. Interesting enough, the trace norm of tca is the square root of the transition probability (fidelity) between w and w,

2

(23)

The estimates are in line with the question how to optimize quantum teleportation. Depending on specific demands, the problem has been addressed

262

by Horodecki et a/.15, Trump et al.ie, Banaczek17, Rehacek et al.18 and others.

2.2. Liiders measurements It is a strong assumption, to suppose Alice could perform rank one measurements. With raising magnitude of degrees of freedom the task become more and more difficult. In the realm of relativistic quantum field theories local measurements with projections of infinite rank are most natural. (Though these systems contain lots of finite dimensional subsystems, one has to find some with sufficiently exposed sets of quantum levels.) Thus, the projection P in the preparing step (18) may be of any rank. Let

p = ^mb)(rkb\

(24)

be an orthogonal decomposition of P into rank one projection operators. Associating EPR-maps

rkb <—> 4a

(25)

to every vector appearing in (24), (18) becomes (P ® 1C)(
tr = < o s£a

(26)

We have to decouple the degrees of freedom coming from the 6-system. To do so, we first convert the maps between vectors in those between (not necessarily normalized) density operators. Then we reduce the right hand side of (26) to the c system. Abbreviating (tco)* by tac, the result is the map

iwi — £tni<ww

(27)

We estimate (26): The norm of the left is smaller than product of the norms off a and if>bc. On the right side orthogonality of the ^fc allows to calculate the norm. We get

Being valid for all vectors from Ha we conclude

The boundedness of the operator allows to extend (27) to a map from the trace class operators on Ha to those of Hc. The extension reads V°sf)s^

(29)

263

with va an arbitrary trace class operator. Estimating the trace of T by (28) one sees |Tca|i < ((f

,(p )

(30)

More general, positive operator valued measurements have been examined by Mor and Horodecki19 and others. 2.3. Distributed measurements In a multipartite system with an even number of subsystems one can distribute the measurements and the entanglement resources over some pairs of subsystems. Let us see this with five subsystems,

The input is an unknown vector <j6a e Ha, the ancillarian vectors are selected from the be- and the de-system,

and the vector of the total system we are starting with is

The channel is triggered by measurements in the ab- and in the erf-system. To see what is going on it suffices to treat rank one measurements. Suppose these measurements prepare, if successful, the vectors

The we get the relation ®e

(33)

and the vector $a is mapped onto (f>e = tea(j)a. Introducing the EPR-maps corresponding to the used vectors s^,


V"* -» s d c ,...,

the factorization property becomes t eo = sed o sdc o sc6 o sba

(34)

264

3. Polar decompositions Coming back to the bipartite case if) £ Tta ® Hb, we shall explore the polar decompositions of the EPR-maps s^a and s^6. As we already know by (12), the positive factors in the polar decompositions must be the square roots of the reduced density operators, wJJ, and w^,, of ij}. Their phase operators are antiunitary partial isometries between the two parts of the bipartite Hilbert space. We call these maps j^a and j!J,6. The first of these antilinear operations maps Ha into Hb, the second Hb into H.a- Standard technique yields the polar decompositions c,ba _ / S

1/l

b\l/2-ba _ ;6a/ , o \ l / 2

— l^V'

JV>

~ J - 0 (Wll>)

'

Sc afc

_ / ,aU/2;a6 506 /. ,6 N l / 2 V> — (UTJ,) J,/, = J,/, (UT/,)

Just as in the linear case, one requires •ab-ba _ /-)a -ba-ab _ /^fe JT/.J,/, — y^,) j^J^ — vv

where QJ, respectively Q^,, denotes the projection operator onto the support space of ui^ and w^, respectively. The unicity of the polar decomposition and (7) yields

a •4,•4,1

ba\* _ -ah

— Jv >

w b _ -ba, a -ah

v —J

One can relate the expectation values of the reduced density operators. Let us prove it as an exercise in antilinearity. We choose A 6 B(T~ia) and B e B(Hb) such that B*$=$A

(38)

Then, neglecting the index i/>,

The trace of the products two antilinear operators, ^1^2, is conjugate complex to the trace of i?2$i- Hence, the expression under consideration is the complex conjugate of

In conclusion it follows Tr^A = Tr^5 from (38).

(39)

265

Another useful observation: Let H'a C Ha be the supporting subspace of a given density operator u>a. The set of all purifications i/> of u>a is in one-to-one correspondence to the set of antilinear isometries from H'a into Hb. Let us further have a look at some relations from which the norm estimates of the teleporting maps will follow. To this end we consider two arbitrary vectors, (p and tp, from H = 'Ha®/Hb- Their polar decompositions, (35), yield ba ab _ -ha

Therefore, the singular values of the operators 6a aft

-ah 60

are equal one to another. The singular values of a Hilbert-Schmidt operator £ are the eigenvalues of the square root of £*£. That way one proves (23) and (22). Notice that for all B e B(Hb) = (V, (1° ® 5)^} = Tr (v^^^)1/2(J^5*Jta)

(42)

As an application let us prove a key statement of the important paper on the mixed state cloning problem by Barnum et al.20 It asserts F(w$,u$ = F(u$,wbv) — > wjwj = u#4

(43)

(See (23) for the definition of F.) It is well know, and easily derived from (42), that the assumption of (43) is satisfied ip and

^>0,

s^°>0

(44)

To say something new, we shall weaken this assumption in requiring only hermiticity instead of positivity. By (7) it means 6a ab _

ba ab

°(f sv> ~ sv> s

ab ba _

ab ba

v ^ ~ *l> f

i
v 4t> /

In the following, starting with (12), we systematically reorder the appearing factors by the the help of (45): , ,a ,a _

— ab ab ab ba _

ab ba ab ba

ab ba ab ba _ ab ba.,ab ba _ _a6 6o = ab_6a °lf; °ip °i/> °ip — °ip si/, S^j si/> — s

and, again by (12), we are done.

266

4. From vectors to Operators on 7ia (g) "Hb With one or two vectors, drawn from the Hilbert space ri of our bipartite system, one can associate operators on it. There are at least two, quite different ways to do so. The first uses the twisted direct product (the twisted Kronecker product) of the EPR maps. In the second one relies on ideas from representation theory, and on an applications of Tomita and Takesaki's theory. All the matter is quite elementary as long as we are within type I factors.

4.1. Twisted direct products The starting point for the following definition are two maps, £6a : Ha ^ Hb,

rjab : rib •-> Ha,

(46) ab

ba

both either linear or antilinear. The twisted direct product, ri ®£ , (with the twisted cross ), is defined by the linear or antilinear extension of a ® <j>b •-> (rjab^ba)(^a ® 4>b) := 7j°V ® tbaa

(47)

The extension has to be linear if both factors are linear maps, and antilinear if both maps are antilinear. Other cases, one map linear and one antilinear, are ill defined. In the admissible cases the Hermitian adjoint can be gained by

(48) Useful is also fo?6®tf°) o (rtf®&) = (r?f C26a) ® (£N2a6)

(49)

Now let tf>, ip £ ria ® rib an ordered pair of vectors. Essentially, there are four twisted products to perform: (50) (51)

The notations are ad hoc ones, with the exception of the last (see below). Because of (48) the Hermitian adjoints of these operators are gained by exchanging the roles of if) and (p. As before, we denote reduced density operators by u> and their supporting projections by Q, decorated, however, with the appropriate indices. To arrive at the polar decompositions we first notice

A^A^ = wj 0 w*.

J^,vJ^ = Ql®Qbf

(52)

267

Reminding the definition (47) and the polar decomposition of the EPRmaps one computes the polar decompositions of the antilinear operators defined above.

4.2. Contact with representation theory There is a representation of B(Ha) with representation space Ha ® Kb associated with the embedding B(Ha) H-> B(W 0 ) lb C B(Ha ® ?4)

Assume that ^ is a cyclic and separating vector, i.e. a GNS-vector for the representation. Equivalently one requires QJ — \a and Q^ = I6. In the spirit of Schrodinger2 one also calls ip completely entangled. With a given second vector,
(55)

for all A € Ha- (55) is a fundamental construct in the theory of Tomita and Takesaki, though, as we are concerned with type I factors, an elementary one: In our case it is not difficult to prove closability of S. We denote the closure of S again by S and write the polar decomposition in standard notation S^ = J^Aj/J, 21

A^= w ;®(^)- 1

(56)

see Haag for an introduction. Having already defined J in (51) as a twisted Kronecker product, we have to show that it coincides with the modular antiunitary operator defined in the theory of Tomita and Takesaki for GNS- vectors V- The most important case is the modular conjugation Jqj, = J^,. Remark that (51) is slightly more general than (55)-. In the former equation if} can be any vector in any bipartite Hilbert space. To prove the assertion we start with a decomposition of unity

to get, by the help of (10), (11) (A® l6>/> =

268

and, finally, J*,vSv,i,(Jw* ® I 6 ) = (1° ® y^)

(57)

Because our starting assumption implies invertibility of o> J , we may rewrite (57) as asserted in (56). References 1. A. Einstein, B. Podolsky, and N. Rosen, Phys.Rev. 47, 777 (1935). 2. The word entanglement, originally Verschranktheit, has been introduced by Schrodinger to call attention to the remarkable properties of superpositions of product vectors in composed quantum systems. See E. Schrodinger, Naturwissenschaften, 35, 807, 823, 844 (1935). 3. A.Peres: Quantum Theory: Concepts and Methods. Kluwer Academic Publ., Dortrecht 1993 4. M. A. Nielsen, I. L. Chuang: Quantum Computation and Quantum Information. Cambridge University Press 2000 5. D. I. Fivel, Phys. Rev. Lett. 74, 835 (1995) 6. A. Uhlmann in Lecture notes in physics, Vol. 539, eds. A. Borowiec, W. Cegla, B. Jancewicz, W. Karwowski. Springer, Berlin 2000 7. A. Uhlmann in Trends in Quantum Mechanics, eds, H.-D. Doebner, S. T. All, M. Keyl, R. F. Werner. World Scientific, Singapore 2000 8. V. P. Belavkin, M. Ohya, Entanglement and compound states in quantum information theory, quant-ph/0004069 9. V. P. Belavkin, Open Sys. & Inf. Dyn., 8, 1 (2001) 10. M. Ohya, Nuovo Cim. 38, 402 (1983) 11. C. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. Wootters, Phys. Rev. Lett., 70, 1895 (1993) 12. G. Brassard, Physica D 120, 43 (1998) 13. R. F. Werner, All Teleportation and Dense Coding Schemes, quantph/0003070 14. S. Albeverio, Shao-M. Fei, Phys. Lett. A 276, 8 (2000) 15. R. Horodecki, M. Horodecki, P. Horodecki, Phys. Lett. A, 222, 21 (1996) 16. C. Trump, D. Brufi, M. Lewenstein, Phys. Lett. A, 279, 1 (2001) 17. K. Banaszek, Phys. Rev. Lett., 86, 1306 (2001) 18. J. Rehacek, Z. Hradil, J. Fiurasek, C. Bruckner, Designing optimal CP maps for quantum teleportation. quant-ph/0105119 19. T. Mor, P. Horodecki, Teleportation via generalized measurements, and conclusive teleportation. Quant-ph/9906039 20. H. Barnum, C. Caves, C. Fuchs, R. Jozsa, and B. Schumacher, Phys. Rev. Lett., 76, 2818 (1996) 21. R. Haag, Local Quantum Physics. Springer Verlag, Berlin, Heidelbi , New York, 1993