coherent states of the q canonical commutation relations for theq deformed canonical commutation relationsa f a g 1 q f g 1 qa g a f forf g in some hilbert space we consider representations generated from a vector satisfyinga f where we show that such a representation exists if and only if 1 moreover for 1 these representations are unitarily equivalent to the fock representation obtained for 0 on the other hand representations obtained for different unit vectors are disjoint we show that the universal c algebra for the relations has a largest proper closed two sided ideal the quotient by this ideal is a naturalq analogue of the cuntz algebra obtained forq 0 we discuss the conjecture that ford this analogue should in fact be equal to the cuntz algebra itself in the limiting casesq 1 we determine all irreducible representations of the relations and characterize those which can be obtained via coherent states

the ito clifford integral abstract we present a theory of non commutative stochastic integration analogous to the ito theory it is shown that wick products of fermi fields define martingales and that stochastic integrals with respect to these are defined for adapted operatorvalued square integrable integrands for square integrable martingales associated with an arbitrary probability gage space a stochastic integral is defined and a doob meyer decomposition for supermartingales obtained

quantum stochastic calculus on full fock modules abstract we develop a quantum stochastic calculus on full fock modules over arbitrary hilbert b b modules we find a calculus of bounded operators where all quantum stochastic integrals are limits of riemann stieltjes sums after having estalished existence and uniqueness of solutions of a large class of quantum stochastic differential equations we find necessary and sufficient conditions for unitarity of a subclass of solutions as an application we find dilations of a conservative cp semigroup quantum dynamical semigroup on b with arbitrary bounded christensen evans generator we point out that in the case b b g the calculus may be interpreted as a calculus on the full fock space tensor initial space g with arbitrary degree of freedom dilating cp semigroups with arbitrary lindblad generator finally we show how a calculus on the boolean fock module reduces to our calculus as a special case this includes a calculus on the boolean fock space

fixed points of functions with max weighted quasi arithmetic mean operator abstract let 0 1 and f be a continuous strictly monotone real valued function the weighted quasi arithmetic mean of two numbers a b is defined by a b f 1 f a 1 f b let a a i j be an n n real matrix and x x 1 x 2 x n t r n we construct a function a 1 a 2 a n a r n r n by j a x max 1 l n x l a l j for all j 1 2 n in this paper we show that a has a unique fixed point x a moreover it can be shown that for each x r n the sequence x a k generated by the following iterative scheme x a 0 x and x a k a x a k 1 for all k 1 converges to the unique fixed point x a besides some properties of the fixed point are derived as an application our results imply that the max weighted quasi arithmetic mean powers of any matrix are always convergent the continuity of the function m n n r r n defined by a x a is proposed as well

quantum levy processes on dual groups we present a theory of quantum non commutative levy processes on dual groups which generalizes the theory of levy processes on bialgebras it follows from a result of n muraki that there exist exactly 5 notions of non commutative positive stochastic independence we show that one can associate a commutative bialgebra with each pair consisting of a dual group and one of the 5 notions of independence this construction is related to a construction of u franz our construction has the advantage that the important case of free independence is included we show that levy processes are given by their generators which are precisely the conditonally positive linear functionals on the dual group

stochastic integration on the cuntz algebra o abstract we develop a theory of non commutative stochastic integration with respect to the creation and annihilation process on the full fock space over l2 r our theory largely parallels the theories of non commutative stochastic ito integration on boson and fermion fock space as developed by r hudson and k r parthasarathy it provides the first example of a non commutative stochastic calculus which does not depend on the quantum mechanical commutation or anticommutation relations but it is based on the theory of reduced free products of c algebras by d voiculescu this theory shows that the creation and annihilation processes on the full fock space over l2 r which generate the cuntz algebra o can be interpreted as a generalized brownian motion we should stress the fact that in contrast to the other theories of stochastic integration our integrals converge in the c norm on o i e uniformly rather than in some state dependent strong operator topology or non commutative l2 norm

addition of freely independent random variables abstract a direct proof is given of voiculescu s addition theorem for freely independent real valued random variables using resolvents of self adjoint operators in contrast to the original proof no assumption is made on the existence of moments above the second

hilbert modules in quantum electro dynamics and quantum probability a physical system of the form with a distinguished state on may be described in a natural way on a hilbert module following the ideas of accardi and lu 1 we apply this possibility to a concrete system consisting of a boson field in the vacuum state coupled to a free electron

an example of a generalized brownian motion we present an example of a generalized brownian motion it is given by creation and annihilation operators on a twisted fock space ofl2 these operators fulfill for a fixed 1 1 the relationsc f c g c g c f f g 1 f g l2 we show that the distribution of these operators with respect to the vacuum expectation is a generalized gaussian distribution in the sense that all moments can be calculated from the second moments with the help of a combinatorial formula we also indicate that our brownian motion is one component of ann dimensional brownian motion which is invariant under the quantum groups u n of woronowicz with v2

on the rough paths approach to non commutative stochastic calculus abstract we study different possibilities to apply the principles of rough paths theory in a non commutative probability setting first we extend previous results obtained by capitaine donati martin and victoir in lyons s original formulation of rough paths theory then we settle the bases of an alternative non commutative integration procedure in the spirit of gubinelli s controlled paths theory and which allows us to revisit the constructions of biane and speicher in the free brownian case new approximation results are also derived from the strategy

the two parameter free unitary segal bargmann transform and its biane gross malliavin identification abstract motivated by a conditional expectation interpretation of the segal bargmann transform we derive the integral kernel for the large n limit of the two parameter segal bargmann hall transform over the unitary group u n and explore its limiting behavior we also extend the notion of circular systems to more general elliptic systems in order to give an alternate construction of our new two parameter free unitary segal bargmann hall transform via a biane gross malliavin type theorem

q gaussian processes non commutative and classical aspects we examine for 1 q 1 q gaussian processes i e families of operators non commutative random variables where the a t fulfill the q commutation relations for some covariance function equipped with the vacuum expectation state we show that there is a q analogue of the gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on q gaussian processes into corresponding and much simpler questions in the underlying hilbert space in particular we use this idea to show that a large class of q gaussian processes possesses a non commutative kind of markov property which ensures that there exist classical versions of these non commutative processes this answers an old question of frisch and bourret fb

rigorous mean field model for coherent potential approximation anderson model with free random variables a model of a randomly disordered system with site diagonal random energy fluctuations is introduced it is an extension of wegner sn orbital model to arbitrary eigenvalue distribution in the electronic level space the new feature is that the random energy values are not assumed to be independent at different sites but free freeness of random variables is an analog of the concept of independence for noncommuting random operators a possible realization is the ensemble of randomly rotated matrices at different lattice sites the one and two particle green functions of the proposed hamiltonian are calculated exactly the eigenstates are extended and the conductivity is novanishing everywhere inside the band the long range behavior and the zero frequency limit of the two particle green function are universal with respect to the eigenvalue distribution in the electronic level space the solutions solve the cpa equation for the one and two particle green function of the corresponding anderson model thus our multisite model is a rigorous mean field model for the single site cpa we show how the lloyd model is included in our model and treat various kinds of noises

fermion ito s formula and stochastic evolutions an ito product formula is proved for stochastic integrals against fermion brownian motion and used to construct unitary processes satisfying stochastic differential equations as in the corresponding boson theory 10 11 these give rise to stochastic dilations of completely positive semigroups

a global selection procedure for polynomial interpolators the drive for efficient methods of testing design ideas and prototypes particularly in engineering has led to a rapid increase in the field of numerical simulation methods using computers such as finite element analysis this increase in popularity has fueled the need for empirical models that interpolate data collected from these simulations we propose a selection algorithm to efficiently explore an appropriate class of polynomial interpolators the storage of polynomial models is made efficient and effective thanks to a special coding finally the last stage of the algorithm returns a model that is no longer an interpolator but having a smaller number of terms is simpler and easier to handle and understand in this way a trade off between accuracy and simplicity of the model is attained

free products of c algebras small spatial free products of c algebras are constructed under certain conditions they have properties similar to those proved by paschke and salinas for the algebras c gi g2 where g1 g2 are discrete groups the freeproduct analogs of noncommutative bernoulli shifts are discussed 0 introduction let k be a field consider the category of unital algebras over k it is well known that this category admits coproducts free products of algebras 2 heuristically the free product of algebras is the algebra generated by them with no relations except for the identification of unit elements if k c the complex numbers and we consider unital algebras we can easily define a operation on the free products let a b be unital c algebras and a b their free product which is a unital algebra the question arises in what ways may one define a pre c norm on a b that extends the norms on a and b guided by analogy with tensor products we expect to have a choice among many pre c norms giving rise to many c free products of a and b one natural norm is 1c i i sup 1it c ih st representation of a b the representations of a b are in 1 1 correspondence with pairs of representations of a and b which act on the same hilbert space let a b be the completion of a b in this norm it is easy to see that this construction defines a coproduct in the category of c algebras and that a b is the biggest free product of a and b analogous to the biggest tensor product a 0 b if g 1 g2 are discrete groups we obtain c g1 c g2 c gi g2 where g1 g2 is the free product group and this is analogous to the relation c g1 0 c g2 c g1 x g2 this paper is motivated by the question is there a smallest c product a b in analogy to the smallest tensor product a b satisfying a relation cr gi g2 cr gi cr g2 received by the editors january 28 1981 and in revised form may 19 1981 1980 mathematics subject classification primary 46l05 secondary 46l55 1 982 american mathematical society 0002 9947 82 00001022 04 25

markov dilations on w algebras abstract in this paper a systematic study of markov dilations is begun for completely positive operators on w algebras which leave a faithful normal state invariant it is shown that a minimal markov dilation preserves important properties of the underlying completely positive operator afterwards some results are proved concerning the construction of dilations which lead to markov dilations for large classes of operators finally some of the ideas developed here are applied to the study of a simple example over the 2 2 matrices

quantum q white noise and a q central limit theorem we establish a connection between the azema martingales and certain quantum stochastic processes with increments satisfyingq commutation relations this leads to a theory ofq white noise onq bialgebras and to a generalization of the fock space representation theorem for white noise on bialgebras in particular quantum azema noise q interpolations between fermion and boson quantum brownian motion and unitary evolutions withq independent multiplicative increments are studied it follows from our results that the azema martingales and theq interpolations are central limits of sums ofq independent identically distributed quantum random variables

quantum ito s formula and stochastic evolutions using only the boson canonical commutation relations and the riemann lebesgue integral we construct a simple theory of stochastic integrals and differentials with respect to the basic field operator processes this leads to a noncommutative ito product formula a realisation of the classical poisson process in fock space which gives a noncommutative central limit theorem the construction of solutions of certain noncommutative stochastic differential equations and finally to the integration of certain irreversible equations of motion governed by semigroups of completely positive maps the classical ito product formula for stochastic differentials with respect to brownian motion and the poisson process is a special case

noncommutative brownian motion in monotone fock space an example of noncommutative brownian motion is constructed on the monotone fock space which is a kind of fock space generated by all the decreasing finite sequences of positive real numbers the probability distribution at time associated to this brownian motion is shown to be the arcsine law normalized to mean 0 and variance t

noncommutative stochastic integration through decoupling abstract we present an abstract approach to noncommutative stochastic integration in the context of a finite von neumann algebra equipped with a normal faithful tracial state with respect to processes with tensor or freely independent increments satisfying a stationarity condition using a decoupling technique we obtain necessary and sufficient conditions for stochastic integrability of l p processes with respect to such integrators we apply the theory to stochastic integration with respect to boson and free brownian motion

quantum stochastic calculus with maximal operator domains quantum stochastic calculus is extended in a new formulation in which its stochastic integrals achieve their natural and maximal domains operator adaptedness conditional expectations and stochastic integrals are all defined simply in terms of the orthogonal projections of the time filtration of fock space together with sections of the adapted gradient operator free from exponential vector domains our stochastic integrals may be satisfactorily composed yielding quantum ito formulas for operator products as sums of stochastic integrals the calculus has seen two reformulations since its discovery one closely related to classical ito calculus the other to noncausal stochastic analysis and malliavin calculus our theory extends both of these approaches and may be viewed as a synthesis of the two the main application given here is existence and uniqueness for the attal meyer equations for implicit definition of quantum stochastic integrals

continuous unital dilations of completely positive semigroups abstract since it began in the 1970s the study of completely positive semigroups has included among its central topics the dilation of a completely positive semigroup to an endomorphism semigroup several authors have proved the existence of dilations but have usually only achieved unital dilations of specific examples while general theorems have tended to produce non unital dilations a unique approach due to jean luc sauvageot leads to a unital dilation in a general setting but leaves unclear the continuity of the dilation semigroup the major purpose of this paper therefore is to further develop sauvageot s theory in order to prove the existence of continuous unital dilations the dilation is characterized by a combinatorial property akin to free independence this property is implicit in some of sauvageot s original calculations but a secondary goal of this paper is to present it as its own object of study which is done in section 4 this work is based on the author s ph d thesis

completely positive maps of the cuntz algebras we construct a covariant functor from the category whose objects are the complex infinite dimensional separable hilbert spaces and whose morphisms are the contractions into the category whose objects are the unitalc algebras and whose morphisms are the completely positive identity preserving maps thec algebras involved are isomorphic to the cuntz algebra o and our functor extends the canonical action of the group of unitaries on the generating hilbert space our result cannot apply to on n

wigner chaos and the fourth moment we prove that a normalized sequence of multiple wigner integrals in a fixed order of free wigner chaos converges in law to the standard semicircular distribution if and only if the corresponding sequence of fourth moments converges to 2 the fourth moment of the semicircular law this extends to the free probabilistic setting some recent results by nualart and peccati on characterizations of central limit theorems in a fixed order of gaussian wiener chaos our proof is combinatorial analyzing the relevant noncrossing partitions that control the moments of the integrals we can also use these techniques to distinguish the first order of chaos from all others in terms of distributions we then use tools from the free malliavin calculus to give quantitative bounds on a distance between different orders of chaos when applied to highly symmetric kernels our results yield a new transfer principle connecting central limit theorems in free wigner chaos to those in gaussian wiener chaos we use this to prove a new free version of an important classical theorem the breuer major theorem

filtered random variables bialgebras and convolutions we introduce the filtered bialgebra which is a multivariate generalization of the unital bialgebra c x x p of polynomials in noncommuting variables x x x x and a projection p p p2 endowed with the coproduct x x 1 1 x x x p p x with p being group like we study the associated convolutions random walks and filtered random variables the gns representations of the limit states lead to filtered fundamental operators which are the ccr fundamental operators on the multiple symmetric fock space h over h l2 r g where g is a separable hilbert space multiplied by appropriate projections the importance of filtered random variables and fundamental operators stems from the fact that by addition and strong limits one obtains from them the main types of noncommutative random variables and fundamental operators respectively regardless of the type of noncommutative independence

quasi free quantum stochastic integrals for the car and ccr abstract a theory of quantum martingales and quantum stochastic integrals in quasi free representations of the car and ccr is presented for the car the results generalize some of those developed in barnett streater and wilde j funct anal 48 1982 172 212 j london math soc 27 1983 373 384 and for the ccr the results contain the standard ito theory of stochastic integration with respect to brownian motion as a special case

quantum random walk for uq su 2 and a new example of quantum noise we derive the quantum version of the central limit theorem for sample sums of q independent q identically distributed quantum variables for q r 1 in particular we consider uq su 2 for which we obtain in the limit a state on the algebra generated by q commuting q oscillators indexed by pairs of real numbers r s where 0 r s giving a new example of non additive quantum noise

quantum and non causal stochastic calculus the quantum stochastic calculus initiated by hudson and parthasarathy and the non causal stochastic calculus originating with the papers of hitsuda and skorohod are two potent extensions of the ito calculus currently enjoying intensive development the former provides a quantum probabilistic extension of schrodinger s equation enabling the construction of a markov process for a quantum dynamical semigroup the latter allows the treatment of stochastic differential equations which involve terms which anticipate the future in this paper the close relationship between these theories is displayed and a noncausal quantum stochastic calculus already in demand from physics is described

a new example of independence and white noise we examine the notion of free independence according to voiculescu this form of independence is used for defining free white noise or process with stationary and freely independent increments we prove a general limit theorem giving the combinatorics of infinitely freely divisible states and thus of free white noises with the help of admissible partitions we realize the free analogues of the wiener process and of the poisson process as processes on the full fock space ofl2