robustness in quantum measurements conditions are formulated under which a representation of an intrinsic c algebra of often quasilocal observables of an infinite system is appropriate to describe measurement type processes such a representation should allow for the description of robust experiments it should be separable and the pointer observable should be in its weak closure if the pointer values are discrete the existence of such a measurement representation can be proven if the pointer can take continuously many values then the existence can only be proven under the additional assumptions of having an asymptotically abelian system or dealing with type i representations in the constructed measurement representations the pointer observable turns out to be classical the structure of the representation suggests that spontaneous symmetry breaking might be a physical explanation of the emergence of the classical pointer

deformations of compact quantum groups via rieffel s quantization it is shown that compact quantum groups containing torus subgroups can be deformed into new compact quantum groups under rieffel s quantization this is applied to showing that the two classes of compact quantum groupskqu andkq studied by levendorkii and soibelman are strict deformation quantization of each other and that the quantum groupsau m have many deformations

representations of the quantum group suq 2 and the little q jacobi polynomials abstract in this paper we study the finite dimensional unitary representations of the quantum group suq 2 then we obtain the peter weyl theorem for suq 2 and the matrix elements of these unitary representations are explicitly expressed in terms of the little q jacobi polynomials which are known as q analogues of orthogonal polynomials using these expressions the orthogonality relations of these polynomials are obtained in terms of the haar measure on the quantum group suq 2

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

the defect of generalized fourier matrices the n n complex hadamard matrices form a real algebraic manifold cn we have cn mn t nun and following tadej and yczkowski we investigate here the computation of the enveloping tangent space t hcn thmn t thnun and notably of its dimension d h dim t hcn called undephased defect of h our main result is an explicit formula for the defect of the fourier matrix fg associated to an arbitrary finite abelian group g zn1 znr we also comment on the general question does the associated quantum permutation group see the defect with a probabilistic speculation involving diaconis shahshahani type variables

tannaka krein duality for compact matrix pseudogroups twisted su n groups the notion of concrete monoidalw category is introduced and investigated a generalization of the tannaka krein duality theorem is proved it leads to new examples of compact matrix pseudogroups among them we have twistedsu n groups denoted bys u n it is shown that the representation theory fors u n is similar to that ofsu n irreducible representations are labeled by young diagrams and formulae for dimensions and multiplicity are the same as in the classical case

quantum deformation of lorentz group a one parameter quantum deformations l 2 ofsl 2 is introduced and investigated an analog of the iwasawa decomposition is proved the compact part of this decomposition coincides withs u 2 whereas the solvable part is identified as a pontryagin dual ofs u 2 it shows thats l 2 is the result of the dual version of drinfeld s double group construction applied tos u 2 the same construction applied to any compact quantum groupgc is discussed in detail in particular the explicit formulae for the haar measures on the pontryagin dualgd ofgc and on the double groupg are given we show that there exists remarkable 1 1 correspondence between representations ofg and bicovariant bimodules tensor bundles overgc the theory of smooth representations ofs l 2 is the same as that ofsl 2 clebsh gordon coefficients are however modified the corresponding tame bicovariant bimodules ons u 2 are classified an application to 4d differential calculus is presented the nonsmooth case is also discussed

stable laws and domains of attraction in free probability theory in this paper we determine the distributional behavior of sums of free in the sense of voiculescu identically distributed inflnitesimal random variables the theory is shown to parallel the classical theory of independent random variables though the limit laws are usually quite difierent our work subsumes all previously known instances of weak convergence of sums of free identically distributed random variables in particular we determine the domains of attraction of stable distributions in the free theory these freely stable distributions are studied in detail in the appendix where their unimodality and duality properties are demonstrated

analytical and numerical aspects of certain nonlinear evolution equations ii numerical nonlinear schrodinger equation abstract various numerical methods are employed in order to approximate the nonlinear schrodinger equation namely i the classical explicit method ii hopscotch method iii implicit explicit method iv crank nicolson implicit scheme v the ablowitz ladik scheme vi the split step fourier method f tappert and vii pseudospectral fourier method fornberg and whitham comparisons between the ablowitz ladik scheme which was developed using notions of the inverse scattering transform and the other utilized schemes are obtained

hopf von neumann algebra bicrossproducts kac algebra bicrossproducts and the classical yang baxter equations two groups g1 g2 are said to be a matched pair if each acts on the space of the other and these actions say obey a certain compatibility condition for every matched pair of locally compact groups g1 g2 we construct an associated coinvolutive hopf von neumann algebra m g1 l g2 by simultaneous cross product and cross coproduct for non trivial these bicrossproduct hopf von neumann algebras are non commutative and non cocommutative if the modules for the actions are also matched then these bicrossproducts are kac algebras in this case we show that the dual kac algebra is of the same form with the roles of g1 g2 and of interchanged examples exist with g1 a simply connected lie group and choices of g2 determined by suitable solutions of the classical yang baxter equations on the complexification of the lie algebra of g1

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

quantum symmetry groups of finite spaces we determine the quantum automorphism groups of finite spaces these are compact matrix quantum groups in the sense of woronowicz

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

a noncommutative de finetti theorem invariance under quantum permutations is equivalent to freeness with amalgamation we show that the classical de finetti theorem has a canonical noncommutative counterpart if we strengthen exchangeability i e invariance of the joint distribution of the random variables under the action of the permutation group to invariance under the action of the quantum permutation group more precisely for an infinite sequence of noncommutative random variables x i i in mathbb n we prove that invariance of the joint distribution of the xi s under quantum permutations is equivalent to the fact that the xi s are identically distributed and free with respect to the conditional expectation onto the tail algebra of the xi s

classical and free infinitely divisible distributions and random matrices we construct a random matrix model for the bijection between classical and free infinitely divisible distributions for every d 1 we associate in a quite natural way to each infinitely divisible distribution a distribution p d on the space of d d hermitian matrices such that p d p d p d the spectral distribution of a random matrix with distribution p d converges in probability to when d tends to it gives among other things a new proof of the almost sure convergence of the spectral distribution of a matrix of the gue and a projection model for the marchenko pastur distribution in an analogous way for every d 1 we associate to each infinitely divisible distribution a distribution l d on the space of complex non hermitian d x d random matrices if is symmetric the symmetrization of the spectral distribution of m d when m d is l d distributed converges in probability to

integration with respect to the haar measure on unitary orthogonal and symplectic group we revisit the work of the first named author and using simpler algebraic arguments we calculate integrals of polynomial functions with respect to the haar measure on the unitary group u d the previous result provided exact formulas only for 2d bigger than the degree of the integrated polynomial and we show that these formulas remain valid for all values of d also we consider the integrals of polynomial functions on the orthogonal group o d and the symplectic group sp d we obtain an exact character expansion and the asymptotic behavior for large d thus we can show the asymptotic freeness of haar distributed orthogonal and symplectic random matrices as well as the convergence of integrals of the itzykson zuber type

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

a quantum mechanics analogy for the nonlinear schrodinger equation in the finite line abstract a quantum mechanics analogy is used to determine the forces acting on and the energies of solitons governed by the nonlinear schrodinger equation in finite intervals with periodic and with homogeneous dirichlet neumann and robin boundary conditions it is shown that the energy densities remain nearly constant for periodic while they undergo large variations for homogeneous boundary conditions the largest variations in the force and energy densities occur for the neumann boundary conditions but for all the boundary conditions considered in this paper the magnitudes of these forces and energies recover their values prior to the interaction of the soliton with the boundary after the soliton rebound process is completed it is also shown that the quantum momentum changes sign but recovers its original value after the collision of the soliton with the boundaries the asymmetry of the robin boundary conditions shows different dynamic behaviour at the left and right boundaries of the finite interval

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

quantum bernstein s theorem and the hyperoctahedral quantum group we study an extension of bernstein theorem to the setting of quantum groups for a d tuple of free identically distributed random variables we consider a problem of preservation of freeness under the transformations from the free orthogonal quantum group for a subset not contained in the hyperoctahedral quantum group we prove that preservation of freeness characterizes wigner s semicircle law we show that freeness is always preserved if the quantum subset is contained in the hyperoctahedral quantum group we provide examples of quantum subsets which show that our result is an extension of results known in the literature we study an extension of bernstein theorem to the setting of quantum groups for a d tuple of free identically distributed random variables we consider a problem of preservation of freeness under the transformations from the free orthogonal quantum group for a subset not contained in the hyperoctahedral quantum group we prove that preservation of freeness characterizes wigner s semicircle law we show that freeness is always preserved if the quantum subset is contained in the hyperoctahedral quantum group we provide examples of quantum subsets which show that our result is an extension of results known in the literature

jauch piron states in w algebraic quantum mechanics a state on a w algebra m is said to fulfill the jauch piron condition if p q 1 for projections p q m implies p q 1 here p q denotes the infimum of p and q in the projection lattice of m the jauch piron condition is a compatibility condition between the algebraic and the lattice theoretic approach for the description of physical systems normal i e weakly continuous states always fulfill the jauch piron condition it is argued that states not fulfilling this condition should be regarded as unphysical it is shown that a state on a finite factor m is singular if and only if projections e f m exist such that e f 1 and e f 0 in particular any pure state on m fulfilling the jauch piron condition is normal which implies that the underlying factor m is of type i furthermore the following result is proved let be a pure jauch piron state on w algebra m with separable predual and without any commutative summand then is normal and a central projection z0 m exists such

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

liberation of orthogonal lie groups abstract we show that under suitable assumptions we have a one to one correspondence between classical groups and free quantum groups in the compact orthogonal case we classify the groups under correspondence with the result that there are exactly 6 of them o n s n h n b n s n b n we investigate the representation theory aspects of the correspondence with the result that for o n s n h n b n this is compatible with the bercovici pata bijection finally we discuss some more general classification problems in the compact orthogonal case notably with the construction of a new quantum group

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

k groups of c algebras deformed by actions of rd abstract we show that the topological k groups of a c algebra deformed by an action of r d are isomorphic to those of the original c a1gebra we do this by exhibiting the deformed algebra as a suitable generalized fixed point algebra this also shows that the deformed algebra is nuclear if the original one is

the forced nonlinear schrodinger equation abstract in many applications the nonlinear schrodinger equation nls occurs with a nonlinear boundary value being specified in these cases one may consider the semi infinite nls as being forced and driven by the boundary value here we numerically study one such case where the nls field at the origin q x 0 t is specified as a boundary value we find that solitons may be created by the forcing we devise a model to predict these numerical results and do so with reasonable quantitative results and excellent qualitative results the model is able to predict the threshold for soliton production fairly accurately

de finetti theorems for easy quantum groups we study sequences of noncommutative random variables which are invariant under quantum transformations coming from an orthogonal quantum group satisfying the easiness condition axiomatized in our previous paper for 10 easy quantum groups we obtain de finetti type theorems characterizing the joint distribution of any infinite quantum invariant sequence in particular we give a new and unified proof of the classical results of de finetti and freedman for the easy groups sn on which is based on the combinatorial theory of cumulants we also recover the free de finetti theorem of kostler and speicher and the characterization of operator valued free semicircular families due to curran we consider also finite sequences and prove an approximation result in the spirit of diaconis and freedman

multiplicative functions on the lattice of non crossing partitions and free convolution the lattice of non crossing partitions was introduced by kreweras kre in 1972 and examined further by poupard pou and edelman edel ede2 in the last time there has been an increased interest in this lattice on one hand from the purely combinatorial point of view siu eds sire bss and on the other hand from a quantum probabilistic point of view gss spel spe2 spe3 we became interested in this lattice when we noticed that it is connected with some new form of convolution for probability measures the free convolution introduced by voiculescu voil voi2 voi3 this connection between the free convolution and the lattice of non crossing partitions is exactly analogous to the connection between the usual convolution and the lattice of all partitions we shall work out this connection in the last part of this work the intention of this article is to show that both the purely combinatorial and the probabilistic point of view can benefit from each other motivated by voiculescu s formula for the r series of free convolution we examine multiplicative functions on the lattice of non crossing partitions and prove our main theorem which is much in the spirit of voiculescu s formula we shall then use this main theorem for deriving some known results on non crossing partitions in a unified way this part of our work is purely combinatorial the free convolution serves only as a motivation for our theorem in the last section we shall switch to the probabilistie side and show how the lattice of non crossing partitions determines the structure of the free convolution voiculescu s formula will then follow as an easy corollary of our main theorem in this respect we shall get a purely combinatorial proof of this formula without any operator algebraic or functional analytic tools