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

an algebraic and geometric approach to non bijective quadratic transformations this work is concerned with an extension of the so called levi civita and kustaanheimo stiefel transformations the extension is achieved along two lines firstly the authors examine the latter transformations in other dimensions than the ones originally considered by on the one hand levi civita and on the other hand by kustaanheimo and stiefel secondly they pass from the compact to the non compact case this leads to quadratic non bijective transformations that they refer to as hurwitz or kustaanheimo stiefel like and quasihurwitz or levi civita like transformations the hurwitz and quasihurwitz transformations are introduced and studied in an algebraic framework which relies on the use of eight dimensional cayley dickson algebras an explicit formulation of the hurwitz and quasihurwitz transformations is also given in terms of clifford algebras the hurwitz transformations are investigated from a geometrical viewpoint indeed they are connected to hopf and pseudohopf fibrations finally some differential aspects of the hurwitz and to a lesser extent quasihurwitz transformations are developed in view of future physical applications

coulomb green s functions in an n dimensional euclidean space the h atom green s function is calculated in an n dimensional euclidean space following the feynman lagrangian formulation the use of generalized polar coordinates allows the expansion of the propagator into partial propagators and the separation of the angular and radial variables the angular part is shown to be a generalized legendre polynomial while the radial part may be transformed in that of the harmonic oscillator the h atom spectrum is given by the poles of the green s function

isometries rigidity and universal covers the goal of this paper is to describe all closed aspherical riemannian manifolds m whose universal covers m have have a nontrivial amount of symmetry by this we mean that isom m is not discrete by the well known theorem of myers steenrod ms this condition is equivalent to isom m 1 m also note that if any cover of m has a nondiscrete isometry group then so does its universal cover m our description of such m is given in theorem 1 2 below the proof of this theorem uses methods from lie theory harmonic maps large scale geometry and the homological theory of transformation groups the condition that m have nondiscrete isometry group appears in a wide variety of problems in geometry since theorem 1 2 provides a taxonomy of such m it can be used to reduce many general problems to verifications of specific examples actually it is not always theorem 1 2 which is applied directly but the main subresults from its proof after explaining in 1 1 the statement of theorem 1 2 we give in 1 2 a number of such applications these range from new characterizations of locally symmetric manifolds to the classification of contractible manifolds covering both compact and finite volume manifolds to a new proof of the nadel frankel theorem in complex geometry

orbit equivalence rigidity and bounded cohomology we establish new results and introduce new methods in the theory of measurable orbit equivalence using bounded cohomology of group representations our rigidity statements hold for a wide uncountable class of groups arising from negative curvature geometry amongst our applications are a measurable mostow type rigidity theorems for products of negatively curved groups b prime factorization results for measure equivalence c superrigidity for orbit equivalence d the first examples of continua of type ii1 equivalence relations with trivial outer automorphism group that are mutually not stably isomorphic

measure homology and singular homology are isometrically isomorphic measure homology is a variation of singular homology designed by thurston in his discussion of simplicial volume zastrow and hansen showed independently that singular homology with real coefficients and measure homology coincide algebraically on the category of cw complexes it is the aim of this paper to prove that this isomorphism is isometric with respect to the l1 seminorm on singular homology and the seminorm on measure homology induced by the total variation this in particular implies that one can calculate the simplicial volume via measure homology as already claimed by thurston for example measure homology can be used to prove gromov s proportionality principle of simplicial volume

on the hydrogen like atom in five dimensional space the authors propose an easy and effective variant of the realization of the dynamical symmetry of hydrogen like atoms in five dimensional space based on the relationship between the schrodinger equation for the isotropic harmonic oscillator in four dimensional complex space and the schrodinger equation for the one electron atom in five dimensional space by the use of the path integral they also establish the relationship between the coulomb green function and the green function for the isotropic harmonic oscillator in four dimensional complex space

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

a converse to dye s theorem every non amenable countable group induces orbit inequivalent ergodic equivalence relations on standard borel probability spaces not every free ergodic measure preserving action of f 2 on a standard borel probability space is orbit equivalent to an action of a countable group on an inverse limit of finite spaces there is a treeable non hyperfinite borel equivalence relation which is not universal for treeable in the b ordering

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

variables separation and superintegrability of the nine dimensional micz kepler problem the nine dimensional micz kepler problem is of recent interest this is a system describing a charged particle moving in the coulomb field plus the field of a so 8 monopole in a nine dimensional space interestingly this problem is equivalent to a 16 dimensional harmonic oscillator via the hurwitz transformation in the present paper we report on the multiseparability a common property of superintegrable systems and the superintegrability of the problem first we show the solvability of the schrodinger equation of the problem by the variables separation method in different coordinates second based on the so 10 symmetry algebra of the system we construct explicitly a set of seventeen invariant operators which are all in the second order of the momentum components satisfying the condition of superintegrability the found number 17 coincides with the prediction of 2n 1 law of maximal superintegrability order in the case n 9 until now this law is accepted to apply only to scalar hamiltonian eigenvalue equations in n dimensional space therefore our results can be treated as evidence that this definition of superintegrability may also apply to some vector equations such as the schrodinger equation for the nine dimensional micz kepler problem the nine dimensional micz kepler problem is of recent interest this is a system describing a charged particle moving in the coulomb field plus the field of a so 8 monopole in a nine dimensional space interestingly this problem is equivalent to a 16 dimensional harmonic oscillator via the hurwitz transformation in the present paper we report on the multiseparability a common property of superintegrable systems and the superintegrability of the problem first we show the solvability of the schrodinger equation of the problem by the variables separation method in different coordinates second based on the so 10 symmetry algebra of the system we construct explicitly a set of seventeen invariant operators which are all in the second order of the momentum components satisfying the condition of superintegrability the found number 17 coincides with the prediction of 2n 1 law of maximal superintegrability order in the case n 9 until now this law is accepted to apply only to scalar hamiltonia

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

measure equivalence rigidity of the mapping class group we show that the mapping class group of a compact orientable surface with higher complexity satisfies the following rigidity in the sense of measure equivalence if the mapping class group is measure equivalent to a discrete group then they are commensurable up to finite kernels moreover we describe all locally compact second countable groups containing a lattice isomorphic to the mapping class group we obtain similar results for finite direct products of mapping class groups

on the so 10 2 dynamical symmetry group of the micz kepler problem in a nine dimensional space we propose an effective algebraic method to investigate the dynamical symmetry of a 9 dimensional micz kepler problem by using the connection between this problem and a 16 dimensional isotropic harmonic oscillator the dynamical symmetry group of the considered problem is found as so 10 2 explicit forms of all group elements are given we also obtain all group elements in the algebraic representation of annihilation and creation operators that are very useful for concrete calculations

quantum invariant families of matrices in free probability we consider self adjoint families of infinite matrices of noncommutative random variables such that the joint distribution of their entries is invariant under conjugation by a free quantum group for the free orthogonal and hyperoctahedral groups we obtain complete characterizations of the invariant families in terms of an operator valued r cyclicity condition this is a surprising contrast with the aldous hoover characterization of jointly exchangeable arrays

integration over quantum permutation groups abstract we find a combinatorial formula for the haar measure of quantum permutation groups this leads to a dynamic formula for laws of diagonal coefficients explaining the poisson free poisson convergence result for characters

gromov s measure equivalence and rigidity of higher rank lattices in this paper the notion of measure equivalence me of countable groups is studied me was introduced by gromov as a measure theoretic analog of quasi isometries all lattices in the same locally compact group are measure equivalent this is one of the motivations for this notion the main result of this paper is me rigidity of higher rank lattices any countable group which is me to a lattice in a simple lie group g of higher rank is commensurable to a lattice in g

a characterization of freeness by invariance under quantum spreading we construct spaces of quantum increasing sequences which give quantum families of maps in the sense of so ltan we then introduce a notion of quantum spreadability for a sequence of noncommutative random variables by requiring their joint distribution to be invariant under taking quantum subsequences our main result is a free analogue of a theorem of ryll nardzewski for an infinite sequence of noncommutative random variables quantum spreadability is equivalent to free independence and identical distribution with respect to a conditional expectation introduction the study of random objects with distributional symmetries is an important subject in modern probability consider a sequence 1 2 of random variables such a sequence is called exchangeable if its distribution is invariant under finite permutations and spreadable if it is invariant under taking subsequences i e if 1 k d l1 lk for all k n and l1 lk in the 1930 s de finetti gave his famous characterization of infinite exchangeable sequences of random variables taking values in 0 1 as conditionally i i d this was extended to variables taking values in a compact hausdorff space by hewitt and savage 6 it was later discovered by ryll nardzewski that de finetti s theorem in fact holds under the apparently weaker condition of spreadability 12 for a comprehensive treatment of distributional symmetries in classical probability the reader is referred to the recent text of kallenberg 8 free probability developed by voiculescu in the 1980 s is based on the notion of free independence for random variables with the highest degree of noncommutativity remarkably there is a deep parallel between the theories of classical and free probability however it is only quite recently that this parallel has been extended to the study of distributional symmetries the breakthrough came with the work of kostler and speicher 10 who discovered that roughly speaking in free probability one should consider quantum distributional symmetries more specifically they defined the notion of quantum exchangeability for a sequence x1 x2 of noncommutative random variables by requiring that for each n n the joint distribution of x1 xn is invariant under the natural action of the quantum permutation group as n of wang 17 they then gave a free analogue of de finetti s theorem for an infinite sequence of noncommutative random variables quantum exchangeability is equivalent to free independence and identical distribution with respect to a conditional expectation 2010 mathematics subject classification 46l54 46l65 60g09

integrable measure equivalence and rigidity of hyperbolic lattices we study rigidity properties of lattices in operatorname isom mathbf h n simeq mathrm so n 1 mathbb r n 3 and of surface groups in operatorname isom mathbf h 2 simeq mathrm sl 2 mathbb r in the context of integrable measure equivalence the results for lattices in operatorname isom mathbf h n n 3 are generalizations of mostow rigidity they include a cocycle version of strong rigidity and an integrable measure equivalence classification despite the lack of mostow rigidity for n 2 we show that cocompact lattices in operatorname isom mathbf h 2 allow a similar integrable measure equivalence classification

generalized kaluza klein monopole quadratic algebras and ladder operators we present a generalized kaluza klein monopole system we solve this quantum superintegrable system on a euclidean taub nut manifold using the separation of variables of the corresponding schr odinger equation in spherical and parabolic coordinates we present the integrals of motion of this system the quadratic algebra generated by these integrals the realization in terms of a deformed oscillator algebra using the daskaloyannis construction and the energy spectrum the structure constants and thecasimir operator are functions not only of the hamiltonian but also of other two integrals commuting with all generators of the quadratic algebra and forming an abelian subalgebra we present another algebraic derivation of the energy spectrum of this system using the factorization method and ladder operators

amenable actions of groups the equivalence between different characterizations of amenable actions of a locally compact group is proved in particular this answers a question raised by r j zimmer in 1977

cout des relations d equivalence et des groupes we study a new dynamical invariant for dicrete groups the cost it is a real number in 1 1 n 1 bounded by the number of generators of the group and it is well behaved with respect to finite index subgroups namely the quantities 1 minus the cost are related by multiplying by the index the cost of every infinite amenable group equals 1 we compute it in some other situations including free products free products with amalgamation and hnn extensions over amenable groups and for direct product situations for instance the cost of the free group on n generators equals n we prove that each possible finite value of the cost is achieved by a finitely generated group it is dynamical because it relies on measure preserving free actions on probability borel spaces in most cases groups have fixed price which implies that two freely acting groups which define the same orbit partition must have the same cost it enables us to distinguish the orbit partitions of probability preserving free actions of free groups of different ranks at the end of the paper we give a mercuriale i e a list of costs of different groups the cost is in fact an invariant of ergodic measure preserving equivalence relations and is defined using graphings a treeing is a measurable way to provide every equivalence class orbit with the structure of a simplicial tree this an example of graphing not every relation admits a treeing we prove that every free action of a cost 1 non amenable group is not treeable but we prove that subrelations of treeable relations are treeable we give examples of relations which cannot be produced by an action of any finitely generated group the cost of a relation which can be decomposed as a direct product is shown to be 1 we define the notion for a relation to be a free product or an hnn extension and compute the cost for the resulting relation from the costs of the building blocks the cost is also an invariant of the pairs von neumann algebra cartan subalgebra

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

the hurwitz transformation nonbilinear version an alternative approach to the hurwitz h transformation reducing euclidean space e8 to euclidean space e5 is developed it is shown how refusal of the bilinearity condition leads to the replacement of the h transformation by its modified nonbilinear left and right version hl and hr transformations the hl hr transformation of the specific type eight dimensional hyperspherical coordinates is investigated the radial coordinate u and the polar angle 2 in this approach transform into u2 and respectively like in the h case action of these transformations on the remaining hyperspherical coordinates unlike in the h case is equivalent to the invariance shutting of one angular triplet and the shutting invariance of another triplet the connection of hl and hr with h is established and the structure of the h transformation itself is revealed on this basis

on the coulomb green function the connection between the green function for the isotropic harmonic oscillator in two dimensional complex space and the coulomb green function is established using the method of path integrals the representations for the coulomb green function are proposed to be useful in concrete applications

cat 1 spaces divergence groups and their commensurators a cat 1 space is a metric geodesic space in which every geodesic triangle is thinner than its associated comparison triangle in the hyperbolic plane b briha gr the cat 1 property is one among many possible generalizations to singular spaces of the notion of negative curvature important examples of cat 1 spaces include riemannian manifolds of sectional curvature k 1 and their convex subsets b g s metric trees and piecewise hyperbolic cell complexes mou da hag be 1 be 2 b br in this paper we establish certain superrigidity results for isometric actions of a group on a cat 1 space in the following two settings a the group is a subgroup of a locally compact group g with comg where g is a sufficiently large discrete subgroup and comg g g g 1 g and share a subgroup of finite index is the commensurator of in g b the group is an irreducible lattice in g n 1g k where each g is a semisimple algebraic group defined over a local field k the issues addressed in this paper are motivated on one hand by earlier work of g a margulis ma dealing with the linear representation theory of where in case a g is a semisimple group and g a lattice and on the other hand by the results of lubotzky mozes and zimmer l m z concerning isometric actions of on trees where comg g is the group of automorphisms of a regular tree and g is a lattice our approach to establishing superrigidity results is based on ergodic theoretic methods developed by margulis ma zi 3 a c b in this context the following notion of boundary of a locally compact group will be useful let b be a standard borel space on which acts by borel automorphisms preserving a finite measure class

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

cocycle superrigidity and bounded cohomology for negatively curved spaces we introduce new techniques to extend superrigidity theory beyond the scope of lie or algebraic groups we construct a cohomological invariant which accounts for and generalizes all known superrigidity results for actions on negatively curved spaces together with a new vanishing result and the machinery of bounded cohomology this enables us to prove a general superrigidity theorem for actions of irreducible lattices on spaces of negative curvature we also prove a cocycle version a la zimmer

l 2 betti numbers of locally compact groups and their cross section equivalence relations we prove that the l betti numbers of a unimodular locally compact group g coincide up to a natural scaling constant with the l betti numbers of the countable equivalence relation induced on a cross section of any essentially free ergodic probability measure preserving action of g as a consequence we obtain that the reduced and un reduced l betti numbers of g agree and that the l betti numbers of a lattice in g equal those of g up to scaling by the covolume of in g we also deduce several vanishing results including the vanishing of the reduced l cohomology for amenable locally compact 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

on the superrigidity of malleable actions with spectral gap some of the most interesting aspects of the dynamics of measure preserving actions of countable groups on probability spaces v rx x are revealed by the study of group measure space von neumann algebras l x x t mvnl and the classification of actions up to orbit equivalence oe i e up to isomorphism of probability spaces carrying the orbits of actions onto each other although one is in von neumann algebras and the other in ergodic theory the two problems are closely related as an oe of actions v rx x a rxy has been shown to implement an algebra isomorphism l x x t l y x a taking l x onto l y and vice versa si dy fm in particular the isomorphism class of l x xi t only depends on the equivalence relation 7 r t gt 1t x g t thus von neumann equivalence vne of group actions requiring isomorphism of their group measure space algebras is weaker than oe since there are exam ples of non oe actions whose associated von neumann algebras are all isomorphic cj1 it is in general strictly weaker on the other hand oe is manifestly weaker than classical conjugacy which for free actions r rx x a rx y requires isomor phism of probability spaces a x p y v satisfying ar a 1 a so in particular r a how much weaker vne and oe can be with respect to conju gacy is best seen in the amenable case where by a celebrated theorem of connes all free ergodic actions of all infinite amenable groups give rise to the same hi factor cl and by dy ow cfw they are undistinguishable under oe as well also any embedding of algebras l x xi t c l y x a with a amenable forces t to be amenable but the non amenable case is extremely complex and for many years progress has been slow mvn2 dy me c2 cw sc even after the discovery of the first rigidity phenomena by connes in von neumann algebras c3 c4 and by zimmer in oe ergodic theory zi z2 this changed dramatically over the last 7 8 years with the advent of a variety of striking rigidity results ful gl g2 mosh h hk p1 p8 see p9 for a survey also sh for a survey on oe rigidity our aim in this paper is to investigate the most extreme such phenomena called strong rigidity which show that for certain classes of source group actions

oe and w superrigidity results for actions by surface braid groups we show that several important normal subgroups of the mapping class group of a surface satisfy the following property any free ergodic probability measure preserving action y x is stablyoe superrigid these include the central quotients of most surface braid groups and most torelli groups and johnson kernels in addition we show that all these groups satisfy the measure equivalence rigidity and we describe all their lattice embeddings using these results in combination with previous results from cik13 we deduce that any free ergodic probabilitymeasure preserving action of almost any surface braid group is stablyw superrigid i e it can be completely reconstructed from its vonneumann algebra

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

on the connection between the hydrogen atom and the harmonic oscillator the continuum case the connection between a three dimensional nonrelativistic hydrogen atom with positive energy and a four dimensional isotropic harmonic oscillator with repulsive potential is established by applying jordan schwinger boson calculus to the algebra of the laplace runge lenz pauli vector the spectrum generating group so 4 2 both for the bound and free states of the three dimensional hydrogen atom arises as a quotient of the group sp 8 r associated to four dimensional isotropic harmonic oscillator with constraint

generalized micz kepler system duality polynomial and deformed oscillator algebras we present the quadratic algebra of the generalized micz kepler system in three dimensional euclidean space e3 and its dual the four dimensional singular oscillator in four dimensional euclidean space e4 we present their realization in terms of a deformed oscillator algebra using the daskaloyannis construction the structure constants are in these cases functions not only of the hamiltonian but also of other integrals commuting with all generators of the quadratic algebra we also present a new algebraic derivation of the energy spectrum of the micz kepler system on the three sphere s3 using a quadratic algebra these results point out also that results and explicit formula for structure functions obtained for quadratic cubic and higher order polynomial algebras in the context of two dimensional superintegrable systems may be applied to superintegrable systems in higher dimensions with and without monopoles

perturbations of certain crossed product algebras by free groups given two von neumann algebras m and n acting on the same hilbert space d m n d m n is defined to be the hausdorff distance between their unit balls the kadison kastler problem asks whether two sufficiently close von neumann algebras are spatially isomorphic in this article we show that if p 0 p 0 is an injective von neumann algebra with a cyclic tracial vector g is a free group is a free action of g on p 0 p 0 and n is a von neumann algebra such that d n p 0 g 7 d n p 0 g 1 7 10 7 then n and p 0 g p 0 g are spatially isomorphic suitable choices of the actions give the first examples of infinite noninjective factors for which this problem has a positive solution

a hidden non abelian monopole in a 16 dimensional isotropic harmonic oscillator we suggest one variant of generalization of the hurwitz transformation by adding seven extra variables that allow an inverse transformation to be obtained using this generalized transformation we establish the connection between the schrodinger equation of a 16 dimensional isotropic harmonic oscillator and that of a nine dimensional hydrogen like atom in the field of a monopole described by a septet of potential vectors in a non abelian model of 28 operators the explicit form of the potential vectors and all the commutation relations of the algebra are given

the intrinsic asymmetry and inhomogeneity of teichmuller space let s sg n be a connected orientable surface of genus g 0 with n 0 punctures let teich s denote the corresponding teichmuller space and let mod s denote the mapping class group of s understanding the analogy of teich s with symmetric spaces is a wellknown theme recall that a complete riemannian manifold x is symmetric if it is symmetric at each point x x the map t 7 t which flips geodesics about x is an isometry symmetric spaces x are homogeneous the isometry group isom x acts transitively on x in his famous paper ro royden studied the possible symmetry and homogeneity of teich s endowed with the teichmuller metric

proper actions of lattices on contractible manifolds every lattice in a connected semi simple lie group g acts properly discontinuously by isometries on the contractible manifold g k k a maximal compact subgroup of g we prove that if acts on a contractible manifold w and if either 1 the action is properly discontinuous or 2 w is equipped with a complete riemannian metric the action is by isometries and with unbounded orbits g is simple with finite center and rank 1 then dimw dimg k

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

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

r cyclic families of matrices in free probability we introduce the concept of r cyclic family of matrices with entries in a noncommutative probability space the definition consists in asking that only the cyclic noncrossing cumulants of the entries of the matrices are allowed to be nonzero let a1 as be an r cyclic family of d d matrices over a noncommutative probability space a we prove a convolution type formula for the explicit computation of the joint distribution of a1 as considered in md a with the natural state in terms of the joint distribution considered in the original space a of the entries of the s matrices several important situations of families of matrices with tractable joint distributions arise by application of this formula moreover let a1 as be a family of d d matrices over a noncommutative probability space a let d md a denote the algebra of scalar diagonal matrices and let c be the subalgebra of md a generated by a1 as d we prove that the r cyclicity of a1 as is equivalent to a property of c namely that c is free from md c with amalgamation over d

orbit equivalence rigidity consider a countable group acting ergodically by measure p reserving transformations on a probability space x and let r be the corresponding orbit equivalence relation on x the following rigidity phenomenon is shown there exist group actions such that the equivalence relation r on x determines the group and the action x uniquely up to finite groups the natural action of sln z on the n torus r n z n for n 2 is one of such examples the interpretation of these results in the context of von neumann algebras provides some support to the conjecture of connes on rigidity of group algebras for groups with property t our rigidity results also give examples of countable equivalence relations of type ii1 which cannot be generated mod 0 by a free action of any group this gives a negative answer to a long standing problem of feldman and moore