pseudo hermitian continuous time quantum walks in this paper we present a model exhibiting a new type of continuous time quantum walk as a quantum mechanical transport process on networks which is described by a non hermitian hamiltonian possessing a real spectrum we call it pseudo hermitian continuous time quantum walk we introduce a method to obtain the probability distribution of walk on any vertex and then study a specific system we observe that the probability distribution on certain vertices increases compared to that of the hermitian case this formalism makes the transport process faster and can be useful for search algorithms

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

investigation of continuous time quantum walk via modules of bose mesner and terwilliger algebras the continuous time quantum walk on the underlying graphs of association schemes has been studied via the algebraic combinatorics structures of association schemes namely semi simple modules of their bose mesner and terwilliger algebras it is shown that the terwilliger algebra stratifies the graph into a d 1 disjoint union of strata which is different from the stratification based on distance except for distance regular graphs in underlying graphs of association schemes the probability amplitudes and average probabilities are given in terms of dual eigenvalues of association schemes such that the amplitudes of observing the continuous time quantum walk on all sites belonging to a given stratum are the same therefore there are at most d 1 different observing probabilities the importance of association scheme in continuous time quantum walk is shown by some worked out examples such as arbitrary finite group association schemes followed by symmetric sn dihedral d2m and cyclic groups at the end it is shown that the highest irreducible representations of terwilliger algebras pave the way to use the spectral distributions method of jafarizadeh and salimi 2005 preprint quant ph 0510174 in studying quantum walk on some rather important graphs called distance regular graphs

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

orthogonal free quantum group factors are strongly 1 bounded abstract we prove that the orthogonal free quantum group factors l f o n are strongly 1 bounded in the sense of jung in particular they are not isomorphic to free group factors this result is obtained by establishing a spectral regularity result for the edge reversing operator on the quantum cayley tree associated to f o n and combining this result with a recent free entropy dimension rank theorem of jung and shlyakhtenko

on the ell 2 betti numbers of universal quantum groups we show that the first ell 2 betti number of the duals of the free unitary quantum groups is one and that all ell 2 betti numbers vanish for the duals of the quantum automorphism groups of full matrix algebras

ccap for universal discrete quantum groups we show that the discrete duals of the free orthogonal quantum groups have the haagerup property and the completely contractive approximation property analogous results hold for the free unitary quantum groups and the quantum automorphism groups of finite dimensional c algebras the proof relies on the monoidal equivalence between free orthogonal quantum groups and su q 2 quantum groups on the construction of a sufficient supply of bounded central functionals for su q 2 quantum groups and on the free product techniques of ricard and xu our results generalize previous work in the kac setting due to brannan on the haagerup property and due to the second author on the ccap

compact matrix pseudogroups the compact matrix pseudogroup is a non commutative compact space endowed with a group structure the precise definition is given and a number of examples is presented among them we have compact group of matrices duals of discrete groups and twisted deformed su n groups the representation theory is developed it turns out that the tensor product of representations depends essentially on their order the existence and the uniqueness of the haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived the form of these relations differs from that in the group case this is due to the fact that the haar measure on pseudogroups is not central in general the corresponding modular properties are discussed the haar measures on the twistedsu 2 group and on the finite matrix pseudogroup are found

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

the boundary of universal discrete quantum groups exactness and factoriality we study the c algebras and von neumann algebras associated with the universal discrete quantum groups they give rise to full prime factors and simple exact c algebras the main tool in our work is the study of an amenable boundary action yielding the akemann ostrand property finally this boundary can be identified with the martin or the poisson boundary of a quantum random walk

symmetries of levy processes on compact quantum groups their markov semigroups and potential theory quantum markov semigroups qms i e strongly continuous semigroups of unital completely positive maps on compact quantum groups are studied we show that translation invariant qmss on the universal or reduced c algebra of a compact quantum group are in one to one correspondence with levy processes on its hopf algebra we use the theory of levy processes on involutive bialgebras to characterize symmetry properties of the associated qms it turns out that the qms is self adjoint resp kms symmetric if and only if the generating functional of the levy process is invariant under the antipode resp the unitary antipode furthermore we study levy processes whose marginal states are invariant under the adjoint action finally some related aspects of the potential theory as dirichlet form derivation and spectral triple are investigated we discuss how the above results apply to compact groups group c algebras of discrete groups free orthogonal quantum groups on and twisted suq 2 quantum group

cut off phenomenon for random walks on free orthogonal quantum groups we give bounds in total variation distance for random walks associated to pure central states on free orthogonal quantum groups as a consequence we prove that the analogue of the uniform plane kac walk on this quantum group has a cut off at n ln n 2 1 cos theta this is the first result of this type for genuine compact quantum groups we also obtain similar results for mixtures of rotations and quantum permutations

free wreath product quantum groups the monoidal category approximation properties and free probability in this paper we find the fusion rules of the free wreath product quantum groups g sn for all compact matrix quantum groups of kac type g and n 4 this is based on a combinatorial description of the intertwiner spaces between certain generating representations of g sn the combinatorial properties of the intertwiner spaces in g sn allow us to obtain several probabilistic applications we prove also the monoidal equivalence between g sn and a compact quantum group whose dual is a discrete quantum subgroup of the free product g suq 2 for some 0 q 1 we obtain as a corollary certain stability results for the operator algebras associated with the free wreath products of quantum groups such as the acpap property and exactness

averaging lemmas without time fourier transform and application to discretized kinetic equations we prove classical averaging lemmas in the l2 framework with the help of the fourier transform in variables x and v but not t this method is then used to study discretized problems arising out of the numerical analysis of kinetic equations

the connes embedding property for quantum group von neumann algebras for a compact quantum group g of kac type we study the existence of a haar trace preserving embedding of the von neumann algebra l g into an ultrapower of the hyperfinite ii 1 factor the connes embedding property for l g we establish a connection between the connes embedding property for l g and the structure of certain quantum subgroups of g and use this to prove that the ii 1 factors l o n and l u n associated to the free orthogonal and free unitary quantum groups have the connes embedding property for all n 4 as an application we deduce that the free entropy dimension of the standard generators of l o n equals 1 for all n 4 we also mention an application of our work to the problem of classifying the quantum subgroups of o n

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

the radial masa in free orthogonal quantum groups abstract we prove that the radial subalgebra in free orthogonal quantum group factors is maximal abelian and mixing and we compute the associated bimodule the proof relies on new properties of the jones wenzl projections and on an estimate of certain scalar products of coefficients of irreducible representations

stochastic aspects of easy quantum groups we consider several orthogonal quantum groups satisfying the easiness assumption axiomatized in our previous paper for each of them we discuss the computation of the asymptotic law of tr uk with respect to the haar measure u being the fundamental representation for the classical groups on sn we recover in this way some well known results of diaconis and shahshahani

amenability and co amenability of algebraic quantum groups ii abstract we continue our study of the concepts of amenability and co amenability for algebraic quantum groups in the sense of a van daele and our investigation of their relationship with nuclearity and injectivity one major tool for our analysis is that every non degenerate representation of the universal c algebra associated to an algebraic quantum group has a unitary generator which may be described in a concrete way

the fusion rules of some free wreath product quantum groups and applications abstract in this article we find the fusion rules of the free wreath product quantum groups s n for any discrete group to do this we describe the spaces of intertwiners between basic corepresentations which allows us to identify the irreducible corepresentations we then apply the knowledge of the fusion rules to prove in most cases several operator algebraic properties of the associated reduced c algebras such as simplicity and uniqueness of the trace we also prove that the associated von neumann algebra is a full type ii 1 factor and that the dual of s n has the haagerup approximation property for all finite groups

on the classification of easy quantum groups abstract in 2009 banica and speicher began to study the compact quantum groups g with s n g o n whose intertwiner spaces are induced by some partitions these so called easy quantum groups have a deep connection to combinatorics we continue their work on classifying these objects by introducing some new examples of easy quantum groups in particular we show that the six easy groups o n s n h n b n s n and b n split into seven cases o n s n h n b n s n b n and b n on the side of free easy quantum groups also we give a complete classification in the half liberated and in the nonhyperoctahedral case

asymptotic infinitesimal freeness with amalgamation for haar quantum unitary random matrices we consider the limiting distribution of u na nu n and bn and more general expressions where an and bn are n n matrices with entries in a unital c algebra mathcal b which have limiting mathcal b valued distributions as n and un is a n n haar distributed quantum unitary random matrix with entries independent from mathcal b under a boundedness assumption we show that u na nu n and bn are asymptotically free with amalgamation over mathcal b moreover this also holds in the stronger infinitesimal sense of belinschi shlyakhtenko

completely bounded multipliers of the fourier algebra of a simple lie group of real rank one the fourier algebra a g of a locally compact group g is the space of matrix coefficients of the regular representation and is the predual of the yon neumann algebra vn g generated by the regular representation of g on l 2 g a multiplier m of a g is a bounded operator on a g given by pointwise multiplication by a function on g also denoted m we say m is a completely bounded multiplier ofa g if the transposed operator on vn g is completely bounded definition below it may be possible to find a net ofa g functions m i ie i say such that mi tends to

comparison of unitary duals of drinfeld doubles and complex semisimple lie groups we determine a substantial part of the unitary representation theory of the drinfeld double of a q deformation of a compact lie group in terms of the complexification of the compact lie group using this we show that the dual of every q deformation of a higher rank compact lie group has central property t we also determine the unitary dual of sl q n mathbb c

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

paths in quantum cayley trees and l 2 cohomology abstract we study existence uniqueness and triviality of path cocycles in the quantum cayley graph of universal discrete quantum groups in the orthogonal case we find that the unique path cocycle is trivial in contrast with the case of free groups where it is proper in the unitary case it is neither bounded nor proper from this geometrical result we deduce the vanishing of the first l 2 betti number of a o i n