admissibility conjecture and kazhdan s property t for quantum groups abstract we give a partial solution to a long standing open problem in the theory of quantum groups namely we prove that all finite dimensional representations of a wide class of locally compact quantum groups factor through matrix quantum groups admissibility conjecture for quantum group representations we use this to study kazhdan s property t for quantum groups with non trivial scaling group strengthening and generalising some of the earlier results obtained by fima kyed and soltan chen and ng daws skalski and viselter and brannan and kerr our main results are i all finite dimensional unitary representations of locally compact quantum groups which are either unimodular or arise through a special bicrossed product construction are admissible ii a generalisation of a theorem of wang which characterises property t in terms of isolation of finite dimensional irreducible representations in the spectrum iii a very short proof of the fact that quantum groups with property t are unimodular iv a generalisation of a quantum version of a theorem of bekka valette proven earlier for quantum groups with trivial scaling group which characterises property t in terms of non existence of almost invariant vectors for weakly mixing representations v a generalisation of a quantum version of kerr pichot theorem proven earlier for quantum groups with trivial scaling group which characterises property t in terms of denseness properties of weakly mixing representations

asymptotic expansions in the clt in free probability we prove edgeworth type expansions for distribution functions of sums of free random variables under minimal moment conditions the proofs are based on the analytic definition of free convolution

generalized differential riccati equation and indefinite stochastic lq control with cross term a stochastic indefinite linear quadratic lq optimal control problem with cross term in a finite time horizon is studied a new generalized differential riccati equation gdre is introduced it is shown that the solvability of the gdre is equivalent to the solvability of the indefinite stochastic lq problem furthermore all of the optimal controls including feedback and open loop can be identified via the solution to the gdre an example is presented to illustrate the theory obtained

stochastic linear quadratic control with conic control constraints on an infinite time horizon this paper is concerned with a stochastic linear quadratic lq control problem in the infinite time horizon where the control is constrained in a given arbitrary closed cone the cost weighting matrices are allowed to be indefinite and the state is scalar valued first the mean square conic stabilizability of the system is defined which is then characterized by a set of simple conditions involving linear matrix inequalities lmis next the issue of well posedness of the underlying optimal lq control which is necessitated by the indefiniteness of the problem is addressed in great detail and necessary and sufficient conditions of the well posedness are presented on the other hand to address the lq optimality two new algebraic equations a la riccati called extended algebraic riccati equations eares along with the notion of their stabilizing solutions are introduced for the first time optimal feedback control as well as the optimal value are explicitly derived in terms of the stabilizing solutions to the eares moreover several cases when the stabilizing solutions do exist are discussed and algorithms of computing the solutions are presented finally numerical examples are provided to illustrate the theoretical results established

around property t for quantum groups we study property t for locally compact quantum groups providing several new characterisations especially related to operator algebraic ergodic theory quantum property t is described in terms of the existence of various kazhdan type pairs and some earlier structural results of kyed chen and ng are strengthened and generalised for second countable discrete unimodular quantum groups with low duals property t is shown to be equivalent to property t 1 1 of bekka and valette this is used to extend to this class of quantum groups classical theorems on typical representations due to kerr and pichot and on connections of property t with spectral gaps due to li and ng and with strong ergodicity of weakly mixing actions on a particular von neumann algebra due to connes and weiss finally we discuss in the appendix equivalent characterisations of the notion of a quantum group morphism with dense image

k amenability of hnn extensions of amenable discrete quantum groups abstract we construct hnn extensions of discrete quantum groups we study their representation theory and we show that an hnn extension of amenable discrete quantum groups is k amenable

free products of compact quantum groups we construct and study compact quantum groups from free products ofc algebras in this connection we discover two mysterious classes of natural compact quantum groups a u m anda o m thea u m s respectivelya o m s are non isomorphic to each other for differentm s and are not obtainable by the ordinary quantization method we also clarify some basic concepts in the theory of compact quantum groups

one to one correspondence between generating functionals and cocycles on quantum groups in presence of symmetry we prove that under a symmetry assumption all cocycles on hopf algebras arise from generating functionals this extends earlier results of r vergnioux and d kyed and has two quantum group applications all quantum levy processes with symmetric generating functionals decompose into a maximal gaussian and purely non gaussian part and the haagerup property for discrete quantum groups is characterized by the existence of an arbitrary proper cocycle

the law of large numbers for free identically distributed random variables let x 1 x 2 be a sequence of free identically distributed random variables with common distribution it was shown by lindsay and pata in a more general context that a sufficient condition for the weak law of large numbers to hold for the sequence x 1 x2 is that formula math we show that this condition is necessary as well as sufficient even though the condition is identical with the corresponding one for commuting independent variables the proof of the result uses the analytical techniques of free convolution theory and it is quite different from the proof of the commutative theorem due to kolmogorov cf feller 1971 an introduction to probability theory and its applications wiley new york

constrained stochastic lqc a tractable approach despite the celebrated success of dynamic programming for optimizing quadratic cost functions over linear systems such an approach is limited by its inability to tractably deal with even simple constraints in this paper we present an alternative approach based on results from robust optimization to solve the stochastic linear quadratic control slqc problem in the unconstrained case the problem may be formulated as a semidefinite optimization problem sdp we show that we can reduce this sdp to optimization of a convex function over a scalar variable followed by matrix multiplication in the current state thus yielding an approach that is amenable to closed loop control and analogous to the riccati equation in our framework we also consider a tight second order cone socp approximation to the sdp that can be solved much more efficiently when the problem has additional constraints both the sdp and socp are tractable in the presence of control and state space constraints moreover compared to the riccati approach they provide much greater control over the stochastic behavior of the cost function when the noise in the system is distributed normally

the haagerup property for locally compact quantum groups the haagerup property for locally compact groups is generalised to the context of locally compact quantum groups with several equivalent characterisations in terms of the unitary representations and positive definite functions established in particular it is shown that a locally compact quantum group g has the haagerup property if and only if its mixing representations are dense in the space of all unitary representations for discrete g we characterise the haagerup property by the existence of a symmetric proper conditionally negative functional on the dual quantum group hat g by the existence of a real proper cocycle on g and further if g is also unimodular we show that the haagerup property is a von neumann property of g this extends results of akemann walter bekka cherix valette and jolissaint to the quantum setting and provides a connection to the recent work of brannan we use these characterisations to show that the haagerup property is preserved under free products of discrete quantum groups

limit laws for random matrices and free products in earlier articles we studied a kind of probability theory in the framework of operator algebras with the tensor product replaced by the free product we prove here that free random variables naturally arise as limits of random matrices and that wigner s semicircle law is a consequence of the central limit theorem for free random variables in this way we obtain a non commutative limit distribution of a general gaussian random matrix as an operator in a certain operator algebra wigner s law being given by the trace of the spectral measure of the selfadjoint component of this operator

dynamic mean variance portfolio selection with no shorting constraints this paper is concerned with mean variance portfolio selection problems in continuous time under the constraint that short selling of stocks is prohibited the problem is formulated as a stochastic optimal linear quadratic lq control problem however this lq problem is not a conventional one in that the control portfolio is constrained to take nonnegative values due to the no shorting restriction and thereby the usual riccati equation approach involving a completion of squares does not apply directly in addition the corresponding hamilton jacobi bellman hjb equation inherently has no smooth solution to tackle these difficulties a continuous function is constructed via two riccati equations and then it is shown that this function is a viscosity solution to the hjb equation solving these riccati equations enables one to explicitly obtain the efficient frontier and efficient investment strategies for the original mean variance problem an example illustrating these results is also presented

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

limit theorems in free probability theory i based on an analytical approach to the definition of multiplicative free convolution on probability measures on the nonnegative line and on the unit circle mathbb t we prove analogs of limit theorems for nonidentically distributed random variables in classical probability theory

superconvergence to the central limit and failure of the cramer theorem for free random variables we show that convergence of the semicircle law in the free central limit theorem for bounded random variables is much better than expected thus the distributions which tend to the semicircle become absolutely continuous in finite time and the densities converge in a very strong sense we also show that the semicircle law is the free convolution of laws which are not semicircular thus proving that cramer s classical result for the normal distribution does not have a free counterpart

a characterization of right coideals of quotient type and its application to classification of poisson boundaries let mathbb g be a co amenable compact quantum group we show that a right coideal of mathbb g is of quotient type if and only if it is the range of a conditional expectation preserving the haar state and is globally invariant under the left action of the dual discrete quantum group we apply this result to the theory of poisson boundaries introduced by izumi for discrete quantum groups and generalize a work of izumi neshveyev tuset on suq n for co amenable compact quantum groups with the commutative fusion rules more precisely we prove that the poisson integral is an isomorphism between the poisson boundary and the right coideal of quotient type by a maximal quantum subgroup of kac type in particular the poisson boundary and the quantum flag manifold are isomorphic for any q deformed classical compact lie group

haagerup property for c algebras and rigidity of c algebras with property t abstract we study the haagerup property for c algebras we first give new examples of c algebras with the haagerup property a nuclear c algebra with a faithful tracial state always has the haagerup property and the permanence of the haagerup property for c algebras is established as a consequence the class of all c algebras with the haagerup property turns out to be quite large we then apply popa s results and show the c algebras with property t have a certain rigidity property unlike the case of von neumann algebras for the reduced group c algebras of groups with relative property t the rigidity property strongly fails in general nevertheless for some groups without nontrivial property t subgroups we show a rigidity property in some cases as examples we prove the reduced group c algebras of the non amenable affine groups of the affine planes have a rigidity property

on superconvergence of sums of free random variables this paper derives sufficient conditions for superconvergence of sums of bounded free random variables and provides an estimate for the rate of superconvergence

stochastic receding horizon control of constrained linear systems with state and control multiplicative noise we develop a receding horizon control approach to stochastic linear systems with control and state multiplicative noise that also contain constraints our receding horizon formulation is based upon an on line optimization that utilizes open loop plus linear feedback and is solved as a semi definite programming problem we also provide a characterization of stability performance and constraint satisfaction properties of the receding horizon controlled system under a specific choice of terminal weight and terminal constraint a simple numerical example is used to illustrate the approach

the full classification of orthogonal easy quantum groups we study easy quantum groups a combinatorial class of orthogonal quantum groups introduced by banica speicher in 2009 we show that there is a countable descending chain of easy quantum groups interpolating between bichon s free wreath product with the permutation group s n and a semi direct product of a permutation action of s n on a free product this reveals a series of new commutation relations interpolating between a free product construction and the tensor product furthermore we prove a dichotomy result saying that every hyperoctahedral easy quantum group is either part of our new interpolating series of quantum groups or belongs to a class of semi direct product quantum groups recently studied by the authors this completes the classification of easy quantum groups we also study combinatorial and operator algebraic aspects of the new interpolating series

convolution semigroups of states convolution semigroups of states on a quantum group form the natural noncommutative analogue of convolution semigroups of probability measures on a locally compact group here we initiate a theory of weakly continuous convolution semigroups of functionals on a c bialgebra the noncommutative counterpart of a locally compact semigroup on locally compact quantum groups we obtain a bijective correspondence between such convolution semigroups and a class of c0 semigroups of maps which we characterise on c bialgebras of discrete type we show that all weakly continuous convolution semigroups of states are automatically norm continuous as an application we deduce a known characterisation of continuous conditionally positive definite hermitian functions on a compact group

gaussian fluctuations of characters of symmetric groups and of young diagrams we study asymptotics of reducible representations of the symmetric groups s q for large q we decompose such a representation as a sum of irreducible components or alternatively young diagrams and we ask what is the character of a randomly chosen component or what is the shape of a randomly chosen young diagram our main result is that for a large class of representations the fluctuations of characters and fluctuations of the shape of the young diagrams are asymptotically gaussian in this way we generalize kerov s central limit theorem the considered class consists of representations for which the characters almost factorize and this class includes for example the left regular representation plancherel measure irreducible representations and tensor representations this class is also closed under induction restriction outer product and tensor product of representations our main tool in the proof is the method of genus expansion well known from the random matrix theory

limit laws for normed and weighted boolean convolutions we prove a central limit theorem and a weak law of large numbers for normed boolean convolutions with weighted components our technical assumptions on normalizations and weights cover many interesting outcomes and both limit laws hold under minimal moment conditions

the central limit theorem for free additive convolution abstract let xn n 1be a sequence of free identically distributed random variables with common distribution then there exist sequences bn n 1and an n 1of positive and real numbers respectively such that sequence of random variables formula converges in distribution to the semicircle law if and only if the function formula is slowly varying in karamata s sense in other words the free domain of attraction of the semicircle law coincides with the classical domain of attraction of the gaussian we prove an analogous result for normal domains of attraction in the sense of linnik

two parameter families of quantum symmetry groups we introduce and study natural two parameter families of quantum groups motivated on one hand by the liberations of classical orthogonal groups and on the other by quantum isometry groups of the duals of the free groups specifically for each pair p q of non negative integers we define and investigate quantum groups o p q b p q s p q and h p q corresponding to respectively orthogonal groups bistochastic groups symmetric groups and hyperoctahedral groups in the first three cases the new quantum groups turn out to be related to the dual free products of free quantum groups studied earlier for h p q the situation is different and we show that h p 0 qiso fp where the latter can be viewed as a liberation of the classical isometry group of the p dimensional torus

levy type characterization of stable laws for free random variables we give a description of stable probability measures relative to free additive convolution the definition of domain of attraction is given and a proof is provided of the noncommutative analogue of levy theorem

k amenability for amalgamated free products of amenable discrete quantum groups abstract the basic notions and results of equivariant kk theory concerning crossed products can be extended to the case of locally compact quantum groups we recall these constructions and prove some useful properties of subgroups and amalgamated free products of discrete quantum groups using these properties and a quantum analogue of the bass serre tree we establish the k amenability of amalgamated free products of amenable discrete quantum groups

haagerup property for c algebras abstract we define the haagerup property for c algebras a and extend this to a notion of relative haagerup property for the inclusion b a where b is a c subalgebra of a let be a discrete group and a normal subgroup of we show that the inclusion a r a r has the relative haagerup property if and only if the quotient group has the haagerup property in particular the inclusion c r c r has the relative haagerup property if and only if has the haagerup property c r has the haagerup property if and only if has the haagerup property we also characterize the haagerup property for in terms of its fourier algebra a

cumulants for random matrices as convolutions on the symmetric group in this paper we show that free cumulants can be naturally seen as the limiting value of cumulants of matrices we define these objects as functions on the symmetric group by some convolution relations involving the generalized moments we state that some characteristic properties of the free cumulants already hold for these cumulants

free products of completely positive maps and spectral sets let f be the free group on n generators and for each word w e f let 1 wi be its length haagerup lo showed that for each 0 r 1 the function h wp r is a positive definite function on f consider the positive definite function q n rinl on h then the free product function cpi cpr on ff z h coincides with h and the result was extended this way by de michele and fig talamanca 7 and by boiejko 4 51 in s it is proved that the free product of the unital positive defined functions ui gi z z is still positive defined on the free product group gi the correspondence between the positive defined functions on a discrete group g and the completely positive maps on the full c algebra c g and the isomorphism c g g2 n c g z c g suggested we consider amalgamated products of unital linear maps on the amalgamated product of a family of unital c algebras over a c subalgebra in section 3 we prove that the amalgamated product of a family of unital completely positive b bimodule maps qi a c is completely positive on the biggest amalgamated free product i a i as an application of our main result in section 2 we obtain some results concerning the dilation of noncommutative families of operators

quantum groups property t and weak mixing for second countable discrete quantum groups and more generally second countable locally compact quantum groups with trivial scaling group we show that property t is equivalent to every weakly mixing unitary representation not having almost invariant vectors this is a generalization of a theorem of bekka and valette from the group setting and was previously established in the case of low dual by daws skalski and viselter our approach uses spectral techniques and is completely different from those of bekka valette and daws skalski viselter by a separate argument we furthermore extend the result to second countable nonunimodular locally compact quantum groups which are shown in particular not to have property t generalizing a theorem of fima from the discrete setting we also obtain quantum group versions of characterizations of property t of kerr and pichot in terms of the baire category theory of weak mixing representations and of connes and weiss in terms of the prevalence of strongly ergodic actions

fusion semi rings arising from quantum groups abstract we study the fusion semirings arising from easy quantum groups we classify all the possible free ones answering a question of t banica and r vergnioux these are exactly the fusion rings of quantum groups without any nontrivial one dimensional representation we then classify the possible groups of one dimensional representations for general easy quantum groups associated to noncrossing partitions as an application we give a unified proof of the haagerup property for a broad class of easy quantum groups recovering as special cases previous results by m brannan and f lemeux we end with some considerations on the description of the full fusion ring in the general case

examples of weakly amenable discrete quantum groups abstract we prove that the free orthogonal and free unitary quantum groups f o n and f u n are weakly amenable and that their cowling haagerup constant is equal to 1 this is achieved by estimating the completely bounded norm of the projections on the coefficients of irreducible representations of their compact duals an argument of monoidal equivalence then allows us to extend this result to quantum automorphism groups of finite spaces and even yields some examples of weakly amenable non unimodular discrete quantum groups with the haagerup property

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

stochastic control for linear systems driven by fractional noises this paper is concerned with optimal control of stochastic linear systems involving fractional brownian motion fbm first as a prerequisite for studying the underlying control problems some new results on stochastic integrals and stochastic differential equations associated with fbm are established then three control models are formulated and studied in the first two models the state is scalar valued and the control is taken as markovian either the problems are completely solved based on a riccati equation for model 1 where the cost is a quadratic functional on state and control variables or optimality is characterized for model 2 where the cost is a power functional the last control model under investigation is a general one where the system involves the stratonovich integral with respect to fbm the state is multidimensional and the admissible controls are not limited to being markovian a new riccati type equation which is a backward stochastic differential equation involving both fbm and normal brownian motion is introduced optimal control and optimal value of the model are explicitly obtained based on the solution to this riccati type equation

stochastic linear quadratic control via semidefinite programming we study stochastic linear quadratic lq optimal control problems over an infinite time horizon allowing the cost matrices to be indefinite we develop a systematic approach based on semidefinite programming sdp a central issue is the stability of the feedback control and we show this can be effectively examined through the complementary duality of the sdp furthermore we establish several implication relations among the sdp complementary duality the generalized riccati equation and the optimality of the lq control problem based on these relations we propose a numerical procedure that provides a thorough treatment of the lq control problem via primal dual sdp it identifies a stabilizing feedback control that is optimal or determines that the problem possesses no optimal solution for the latter case we develop an epsilon approximation scheme that is asymptotically optimal

relationship between backward stochastic differential equations and stochastic controls a linear quadratic approach it is well known that backward stochastic differential equations bsdes stem from the study on the pontryagin type maximum principle for optimal stochastic controls a solution of a bsde hits a given terminal value which is a random variable by virtue of an it additional martingale term and an indefinite initial state this paper attempts to explore the relationship between bsdes and stochastic controls by interpreting bsdes as some stochastic optimal control problems more specifically associated with a bsde a new stochastic control problem is introduced with the same dynamics but a definite given initial state the martingale term in the original bsde is regarded as the control and the objective is to minimize the second moment of the difference between the terminal state and the terminal value given in the bsde this problem is solved in a closed form by the stochastic linear quadratic lq theory developed recently the general result is then applied to the black scholes model where an optimal mean variance hedging portfolio is obtained explicitly in terms of the option price finally a modified model is investigated where the difference between the state and the expectation of the given terminal value at any time is taken into account

indefinite stochastic linear quadratic control and generalized differential riccati equation a stochastic linear quadratic lq control problem is indefinite when the cost weighting matrices for the state and the control are allowed to be indefinite indefinite stochastic lq theory has been extensively developed and has found interesting applications in finance however there remains an outstanding open problem which is to identify an appropriate riccati type equation whose solvability is it equivalent to the solvability of the indefinite stochastic lq problem this paper solves this open problem for lq control in a finite time horizon a new type of differential riccati equation called the generalized differential riccati equation is introduced which involves algebraic equality inequality constraints and a matrix pseudoinverse it is then shown that the solvability of the generalized riccati equation is not only sufficient but also necessary for the well posedness of the indefinite lq problem and the existence of optimal feedback open loop controls moreover all of the optimal controls can be identified via the solution to the riccati equation an example is presented to illustrate the theory developed

asymptotic abelianness weak mixing and property t let g be a second countable locally compact group and h a closed subgroup we characterize the lack of kazhdan s property t for the pair g h by the genericity of g actions on the hyperfinite ii1 factor with a certain asymptotic abelianness property relative to h as well as by the genericity of measure preserving g actions on a nonatomic standard probability space that are weakly mixing for h the latter furnishes a definitive generalization of a classical theorem of halmos for single automorphisms and strengthens a recent result of glasner thouvenot and weiss on generic ergodicity we also establish a weak mixing version of glasner and weiss s characterization of property t for discrete g in terms of the invariant state space of a bernoulli shift and show that on the car algebra a type of norm asymptotic abelianness is generic for g actions when g is discrete and admits a nontorsion abelian quotient

addition of certain non commuting random variables soit g le groupe libre non commutatif sur deux generateurs et soit u j j 1 2 les unites dans l 2 g correspondant a la translation a gauche par les generateurs et on considere l 2 g la fonction g g e soient x j des operateurs de la forme x j j u j les operateurs x j peuvent etre consideres comme des variables aleatoires de moments ou de facon equivalente avec des distributions donnees par les fonctionnelles analytiques j ou j f avec ces conventions la distribution de x 1 x 2 depend seulement des distributions de x 1 et x 2 on explique cette relation

the q t gaussian process the q t fock space fq t h introduced in this paper is a deformation of the q fock space of bozejko and speicher the corresponding creation and annihilation operators now satisfy the commutation relation aq t f aq t g qaq t g aq t f f g htn turn mathjax off a defining relation of the chakrabarti jagannathan deformed quantum oscillator algebra the moments of the deformed gaussian element sq t h aq t h aq t h are encoded by the joint statistics of crossings and nestings in pair partitions the q 0 t specialization yields a natural single parameter deformation of the full boltzmann fock space of free probability with the corresponding semicircular measure variously encoded via the rogers ramanujan continued fraction the t airy function the t catalan numbers of carlitz riordan and the first order statistics of the reduced wigner process

local limit theorems in free probability theory in this paper we study the superconvergence phenomenon in the free central limit theorem for identically distributed unbounded summands we prove not only the uniform convergence of the densities to the semicircular density but also their l p convergence to the same limit for p 1 2 moreover an entropic central limit theorem is obtained as a consequence of the above results

the baum connes conjecture for free orthogonal quantum groups we prove an analogue of the baum connes conjecture for free orthogonal quantum groups more precisely we show that these quantum groups have a element and that 1 it follows that free orthogonal quantum groups are k amenable we compute explicitly their k theory and deduce in the unimodular case that the corresponding reduced c algebras do not contain nontrivial idempotents our approach is based on the reformulation of the baum connes conjecture by meyer and nest using the language of triangulated categories an important ingredient is the theory of monoidal equivalence of compact quantum groups developed by bichon de rijdt and vaes this allows us to study the problem in terms of the quantum group suq 2 the crucial part of the argument is a detailed analysis of the equivariant kasparov theory of the standard podle

a cohomological description of property t for quantum groups abstract we prove a delorme guichardet type theorem for discrete quantum groups expressing property t of the quantum group in question in terms of its first cohomology groups as an application we show that the first l 2 betti number of a discrete property t quantum group vanishes

stochastic linear quadratic regulators with indefinite control weight costs ii in part i of this paper s chen x li and x zhou siam j control optim 36 1998 pp 1685 1702 an optimization model of stochastic linear quadratic regulators lqrs with indefinite control cost weighting matrices is proposed and studied in this sequel the problem of solving lqr models with system diffusions dependent on both state and control variables which is left open in part i is tackled first the solvability of the associated stochastic riccati equations sres is studied in the normal case namely all the state and control weighting matrices and the terminal matrix in the cost functional are nonnegative definite with at least one positive definite which in turn leads to an optimal state feedback control of the lqr problem in the general indefinite case the problem is decomposed into two optimal lqr problems one with a forward dynamics and the other with a backward dynamics the well posedness and solvability of the original lqr problem are then obtained by solving these two subproblems and an optimal control is explicitly constructed examples are presented to illustrate the results

closed quantum subgroups of locally compact quantum groups abstract we investigate the fundamental concept of a closed quantum subgroup of a locally compact quantum group two definitions one due to s vaes and one due to s l woronowicz are analyzed and relations between them discussed among many reformulations we prove that the former definition can be phrased in terms of quasi equivalence of representations of quantum groups while the latter can be related to an old definition of podle from the theory of compact quantum groups the cases of classical groups duals of classical groups compact and discrete quantum groups are singled out and equivalence of the two definitions is proved in the relevant context a deep relationship with the quantum group generalization of the herz restriction theorem from classical harmonic analysis is also established in particular in the course of our analysis we give a new proof of the herz restriction theorem

indefinite stochastic riccati equations this paper is concerned with stochastic riccati equations sres which are a class of matrix valued nonlinear backward stochastic differential equations bsdes the sres under consideration are in general indefinite in the sense that certain parameter matrices are indefinite this kind of equations arises from the stochastic linear quadratic lq optimal control problem with random coefficients and indefinite state and control weighting costs the latter having profound implications in both theory and applications while the solvability of the sres is the key to solving the indefinite stochastic lq control it remains in general an extremely difficult open problem this paper attempts to solve the problem of existence and uniqueness of solutions to the indefinite sres for a number of special yet important cases

topological generation and matrix models for quantum reflection groups abstract we prove that the von neumann algebras of quantum permutation groups and quantum reflection groups have the connes embedding property we do this by establishing several new topological generation results for the quantum permutation groups s n and the quantum reflection groups h n s we use these results to show that these quantum groups admit sufficiently many matrix models in particular all of these quantum groups have residually finite discrete duals and are in particular hyperlinear and certain flat matrix models for s n are inner faithful

quantum stochastic convolution cocycles ii schurmann s theory of quantum levy processes and more generally the theory of quantum stochastic convolution cocycles is extended to the topological context of compact quantum groups and operator space coalgebras quantum stochastic convolution cocycles on a c hyperbialgebra which are markov regular completely positive and contractive are shown to satisfy coalgebraic quantum stochastic differential equations with completely bounded coefficients and the structure of their stochastic generators is obtained automatic complete boundedness of a class of derivations is established leading to a characterisation of the stochastic generators of homomorphic convolution cocycles on a c bialgebra two tentative definitions of quantum levy process on a compact quantum group are given and with respect to both of these it is shown that an equivalent process on fock space may be reconstructed from the generator of the quantum levy process in the examples presented connection to the algebraic theory is emphasised by a focus on full compact quantum groups

the haagerup approximation property for von neumann algebras via quantum markov semigroups and dirichlet forms the haagerup approximation property for a von neumann algebra equipped with a faithful normal state is shown to imply existence of unital preserving and kms symmetric approximating maps this is used to obtain a characterisation of the haagerup approximation property via quantum markov semigroups extending the tracial case result due to jolissaint and martin and further via quantum dirichlet forms

e theory and kk theory for groups which act properly and isometrically on hilbert space a good deal of research in c algebra k theory in recent years has been devoted to the baum connes conjecture 3 which proposes a formula for the k theory of group c algebras that blends group homology with the representation theory of compact subgroups the conjecture has brought c algebra theory into close contact with manifold theory through its obvious similarity to the borel conjecture of surgery theory 9 31 and its links with the theory of positive scalar curvature 27 in addition there are points of contact with harmonic analysis particularly the tempered representation theory of semisimple groups although the proper relation between the baum connes conjecture and representation theory is not well understood the conjecture is most easily formulated for groups which are discrete and torsion free for such a group g there is a natural homomorphism

a noncommutative extended de finetti theorem abstract the extended de finetti theorem characterizes exchangeable infinite sequences of random variables as conditionally i i d and shows that the apparently weaker distributional symmetry of spreadability is equivalent to exchangeability our main result is a noncommutative version of this theorem in contrast to the classical result of ryll nardzewski exchangeability turns out to be stronger than spreadability for infinite sequences of noncommutative random variables out of our investigations emerges noncommutative conditional independence in terms of a von neumann algebraic structure closely related to popa s notion of commuting squares and kummerer s generalized bernoulli shifts our main result is applicable to classical probability quantum probability in particular free probability braid group representations and jones subfactors

on the multiplication of free n tuples of noncommutative random variables let a1 a n b 1 b n be random variables in a noncommutative probability space such that a1 a n is free from b1 b n we show how the joint distribution of the n tuple a1b1 a n b n can be described in terms of the joint distributions of a1 a n a nd b 1 b n by using the combinatorics of the n dimensional r transform we point out a few applications that can be easily derived from our result concerning the left and right translation with a semicircular element see sections 1 6 1 10 and the compression with a projection see sections 1 11 1 14 of an n tuple of noncommutative random variables a different approach to two of these applications is presented by dan voiculescu in an appendix to the paper introduction the theory of free random variables was developed in a se quence of papers of d voiculescu see 17 or the recent survey in 16 as an instrument for approaching free products of operator algebras its particular aspect addressed in the present paper is the one concerning the addition and multiplica tion of free random variables as shown by voiculescu in 13 14 a powerful method in the study of these operations is the use of transforms that convert them respectively into addition and multiplication of complex analytic func tions or in an algebraic framework of formal power series in an indeterminate z the precise definitions of these transforms called r transform for the addition problem and s transform for the multiplication problem will be reviewed in the sections 1 2 1 3 below in the present paper we are pursuing a combination of two ideas that have appeared recently in the study of the r and s transforms the first idea is that the connection between the r and the s transform is closer than one might suspect at first glance a way of making this precise was pointed out in our paper 6 in the form of the equation s r for a distribution with nonvan ishing mean and where is a combinatorial object with a precise significance the fourier transform for multiplicative functions on noncrossing partitions a byproduct of our result in 6 is the remark that the multiplication of free

convolution operations arising from vandermonde matrices different types of convolution operations involving large vandermonde matrices are considered the convolutions parallel those of large gaussian matrices and additive and multiplicative free convolution and include additive and multiplicative convolution of vandermonde matrices and deterministic diagonal matrices and cases where two independent vandermonde matrices are involved it is also shown that the convergence of any combination of vandermonde matrices is almost sure the convolutions are divided into two types those which depend on the phase distribution of the vandermonde matrices and those which depend only on the spectra of the matrices a general criterion is presented to find which type applies for any given convolution a simulation is presented verifying the results implementations of the presented convolutions are provided and discussed the implementation is based on the technique of fourier motzkin elimination and is quite general as it can be applied to virtually any combination of vandermonde matrices connections with related matrices such as toeplitz and hankel matrices are also discussed

two state free brownian motions in a two state free probability space a we define an algebraic two state free brownian motion to be a process with two state freely independent increments whose two state free cumulant generating function r z is quadratic note that a priori the distribution of the process with respect to the second state is arbitrary we show however that if a is a von neumann algebra the states are normal and is faithful then there is only a one parameter family of such processes moreover with the exception of the actual free brownian motion corresponding to these processes only exist for finite time