on tracial approximation let c be a class of unital c algebras the class tac of c algebras which can be tracially approximated in the egorov like sense first considered by lin by the c algebras in c is studied lin considered the case that c consists of finite dimensional c algebras or the tensor products of such with c 0 1 in particular the question is considered whether for any simple separable a tac there is a c algebra b which is a simple inductive limit of certain basic homogeneous c algebras together with c algebras in c such that the elliott invariant of a is isomorphic to the elliott invariant of b an interesting case of this question is answered in the final part of the paper the question is also considered which properties of c algebras are inherited by tracial approximation results of this kind are obtained which are used in the proof of the main theorem of the paper and also in the proof of the classification theorem of the second author given in z niu a classification of tracially approximately splitting tree algebra in preparation and z niu a classification of certain tracially approximately subhomogeneous c algebras phd thesis university of toronto 2005 which also uses the main result of the present paper

quasidiagonal morphisms and homotopy abstract adapting certain methods of voiculescu we show that the quasidiagonality of c algebras is invariant under completely positive deformations coming from asymptotic morphisms we define a notion of quasidiagonal morphism the quasidiagonality of morphisms from exact separable c algebras is shown to be invariant under completely positive deformations

independence properties in subalgebras of ultraproduct ii1 factors let mn be a sequence of finite factors with dim mn and denote by m mn their ultraproduct over a free ultrafilter we prove that if q m is either an ultraproduct q qn of subalgebras qn mn with qn mnqn mn n or the centralizer q b m of a separable amenable subalgebra b m then for any separable subspace x m q m there exists a diffuse abelian von neumann subalgebra in q which is free independent to x relative to q m some related independence properties for subalgebras in ultraproduct ii1 factors are also discussed

statistical analysis of multiple sociometric relations abstract loglinear models are adapted for the analysis of multivariate social networks a set of sociometric relations among a group of actors models that focus on the similarities and differences between the relations and models that concentrate on individual actors are discussed this approach allows for the partitioning of the actors into blocks or subgroups some ideas for combining these models are described and the various models and computational methods are applied to the analysis of data for a corporate interlock network of the 25 largest organizations in minneapolis st paul and for a classic network of 18 monks in a cloister the computational techniques all involve variations on the standard iterative proportional fitting algorithm used extensively in the analysis of multidimensional contingency tables

groups associated to ii1 factors we extend recent work of the first named author constructing a natural hom semigroup associated to any pair of ii1 factors this semigroup always satisfies cancelation hence embeds into its grothendieck group when the target is an ultraproduct of a mcduff factor e g r this grothendieck group turns out to carry a natural vector space structure in fact it is a banach space with natural actions of outer automorphism groups

c stability of discrete groups abstract a group may be considered c stable if almost representations of the group in a c algebra are always close to actual representations we initiate a systematic study of which discrete groups are c stable or only stable with respect to some subclass of c algebras e g finite dimensional c algebras we provide criteria and invariants for stability of groups and this allows us to completely determine stability non stability of crystallographic groups surface groups virtually free groups and certain baumslag solitar groups we also show that among the non trivial finitely generated torsion free 2 step nilpotent groups the only c stable group is z

nonnuclear subalgebras of af algebras we show that any non type i separable unital af algebra b can be modeled from inside by a nonnuclear c algebra and from outside by a nonexact c algebra more precisely there exist unital separable quasidiagonal c algebras a b c of real rank zero stable rank one such that a is nonnuclear c is nonexact and both a and c are asymptotically homotopy equivalent to b in particular a b and c have the same ordered k theory groups hence isomorphic ideal lattices and a and b have affinely homeomorphic trace spaces

concavification of free entropy abstract we introduce a modification of voiculescu s free entropy which coincides with the lim inf variant of voiculescu s free entropy on extremal states but is a concave upper semi continuous function on the trace state space we also extend the orbital free entropy of hiai et al 2009 8 to non hyperfinite multivariables and prove freeness in the case of additivity of voiculescu s entropy or vanishing of our extended orbital entropy

numerical range for random matrices abstract we analyze the numerical range of high dimensional random matrices obtaining limit results and corresponding quantitative estimates in the non limit case for a large class of random matrices their numerical range is shown to converge to a disc in particular numerical range of complex ginibre matrix almost surely converges to the disk of radius 2 since the spectrum of non hermitian random matrices from the ginibre ensemble lives asymptotically in a neighborhood of the unit disk it follows that the outer belt of width 2 1 containing no eigenvalues can be seen as a quantification the non normality of the complex ginibre random matrix we also show that the numerical range of upper triangular gaussian matrices converges to the same disk of radius 2 while all eigenvalues are equal to zero and we prove that the operator norm of such matrices converges to 2 e

fluctuations of matrix entries of regular functions of wigner matrices we study the fluctuations of the matrix entries of regular functions of wigner random matrices in the limit when the matrix size goes to infinity in the case of the gaussian ensembles goe and gue this problem was considered by a lytova and l pastur j stat phys 134 147 159 2009 our results are valid provided the off diagonal matrix entries have finite fourth moment the diagonal matrix entries have finite second moment and the test functions have four continuous derivatives in a neighborhood of the support of the wigner semicircle law moreover if the marginal distributions satisfy the poincare inequality our results are valid for lipschitz continuous test functions

on the reduction criterion for random quantum states in this paper we study the reduction criterion for detecting entanglement of large dimensional bipartite quantum systems we first obtain an explicit formula for the moments of a random quantum state to which the reduction criterion has been applied we show that the empirical eigenvalue distribution of this random matrix converges strongly to a limit that we compute in three different asymptotic regimes we then employ tools from free probability theory to study the asymptotic positivity of the reduction operators finally we compare the reduction criterion with other entanglement criteria via thresholds

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

products of independent elliptic random matrices for fixed m 1 we study the product of m independent n times n elliptic random matrices as n tends to infinity our main result shows that the empirical spectral distribution of the product converges with probability 1 to the m th power of the circular law regardless of the joint distribution of the mirror entries in each matrix this leads to a new kind of universality phenomenon the limit law for the product of independent random matrices is independent of the limit laws for the individual matrices themselves our result also generalizes earlier results of gotze tikhomirov on the asymptotic spectrum of products of independent random matrices available at http arxiv org abs 1012 2710 and o rourke soshnikov j probab 16 81 2219 2245 2011 concerning the product of independent iid random matrices

on sofic actions and equivalence relations abstract the notion of sofic equivalence relation was introduced by gabor elek and gabor lippner their technics employ some graph theory here we define this notion in a more operator algebraic context starting from connes embedding problem and prove the equivalence of these two definitions we introduce a notion of sofic action for an arbitrary group and prove that an amalgamated product of sofic actions over amenable groups is again sofic we also prove that an amalgamated product of sofic groups over an amenable subgroup is again sofic

on groups with quasidiagonal c algebras abstract we examine the question of quasidiagonality for c algebras of discrete amenable groups from a variety of angles we give a quantitative version of rosenberg s theorem via paradoxical decompositions and a characterization of quasidiagonality for group c algebras in terms of embeddability of the groups we consider several notable examples of groups such as topological full groups associated with cantor minimal systems and abels celebrated example of a finitely presented solvable group that is not residually finite and show that they have quasidiagonal c algebras finally we study strong quasidiagonality for group c algebras exhibiting classes of amenable groups with and without strongly quasidiagonal c algebras

perspective on the lagrange jacobi mesh this paper presents a unified treatment of the kinetic energy matrix elements related to a number of lagrange functions associated with the lagrange jacobi mesh the matrix elements can be readily modified for application to problems requiring eigenfunction expansion with lagrange legendre lagrange chebyshev lagrange gegenbauer as well as the lagrange jacobi functions the applicability of and the accuracy attainable with the matrix elements is demonstrated with the solution to the schrodinger equation for confining trigonometric poschl teller potentials the results obtained are within machine accuracy when appropriate choices of the basis functions are used

amenability tubularity and embeddings into mathcal r omega suppose m is a tracial von neumann algebra embeddable into mathcal r omega the ultraproduct of the hyperfinite ii1 factor and x is an n tuple of selfadjoint generators for m denote by x m k the microstate space of x of order m k we say that x is tubular if for any e 0 there exist m in mathbb n and 0 such that if x 1 ldots x n y 1 ldots y n in gamma x m k gamma then there exists a k k unitary u satisfying ux iu y i 2 epsilon for each 1 i n we show that the following conditions are equivalent m is amenable i e injective x is tubular any two embeddings of m into mathcal r omega are conjugate by a unitary u in mathcal r omega

a new application of random matrices ext c _ mathrm red f_2 is not a group in the process of developing the theory of free probability and free entropy voiculescu introduced in 1991 a random matrix model for a free semicircular system since then random matrices have played a key role in von neumann algebra theory cf v8 v9 the main result of this paper is the following extension of voiculescui s random matrix result let x n 1 x n r be a system of r stochastically independent n i n gaussian self adjoint random matrices as in voiculescui s random matrix paper v4 and let x1 xr be a semi circular system in a c probability space then for every polynomial p in r noncommuting variables lim niaeia p x n 1 o x n r o p x1 xr for almost all o in the underlying probability space we use the result to show that the ext invariant for the reduced c algebra of the free group on 2 generators is not a group but only a semi group this problem has been open since anderson in 1978 found the first example of a c algebra a for which ext a is not a group

the unexplained accuracy of the lagrange mesh method the lagrange mesh method is an approximate variational calculation which resembles a mesh calculation because of the use of a gauss quadrature in order to analyze its accuracy four different lagrange mesh calculations based on the zeros of laguerre polynomials are compared with exact variational calculations based on the corresponding laguerre basis the comparison is performed for three solvable radial potentials the morse harmonic oscillator and coulomb potentials the results show that the accuracies of the energies obtained for different partial waves with the different mesh approximations are very close to the variational accuracy even in the presence of the centrifugal singularity the same property holds for the approximate wave functions this striking accuracy remains unexplained

stochastic blockmodels for directed graphs abstract holland and leinhardt 1981 proposed the p 1 model for the analysis of binary directed graph data in network studies such a model provides information about the attractiveness and expansiveness of the individual nodes in the network as well as the tendency of a pair of nodes to reciprocate relational ties when the nodes are a priori partitioned into subgroups based on attributes such as race and sex the density of ties from one subgroup to another can differ considerably from that relating another pair of subgroups thus creating a situation called blocking in social networks the p 1 model completely ignores this extra piece of information and is therefore unable to explain the block structure blockmodels that are simple extensions of the p 1 model are proposed specifically for such data an iterative scaling algorithm is presented for fitting the model parameters by maximum likelihood the methodology is illustrated in detail on two empirical examples

convex sets associated to c algebras abstract for a separable unital c algebra a and a separable mcduff ii 1 factor m we show that the space h om w a m of weak approximate unitary equivalence classes of unital homomorphisms a m may be considered as a closed bounded convex subset of a separable banach space a variation on n brown s convex structure h om n r u many separable unital c algebras including all separable unital nuclear c algebras have the property that for any mcduff ii 1 factor m h om w a m is affinely homeomorphic to the trace space of a in general h om w a m and the trace space of a do not share the same data several examples are provided we characterize extreme points of h om w a m in many cases and we give two different conditions one necessary and the other sufficient for extremality in general the universality of c f is reflected in the fact that for any unital separable a h om w a m may be embedded as a face in h om w c f m we also extend brown s construction to apply more generally to h om a m u

on kirchberg s embedding problem kirchberg s embedding problem kep asks whether every separable c algebra embeds into an ultrapower of the cuntz algebra o 2 o 2 in this paper we use model theory to show that this conjecture is equivalent to a local approximate nuclearity condition that we call the existence of good nuclear witnesses in order to prove this result we study general properties of existentially closed c algebras along the way we establish a connection between existentially closed c algebras the weak expectation property of lance and the local lifting property of kirchberg the paper concludes with a discussion of the model theory of o 2 o 2 several results in this last section are proven using some technical results concerning tubular embeddings a notion first introduced by jung for studying embeddings of tracial von neumann algebras into the ultrapower of the hyperfinite ii 1 factor

quadratic forms in unitary operators abstract let u 1 u n be unitary matrices on l 2 denote by the matrix a defined by a i i j j k 1 n u k i j u k i j acting as a bounded operator on l 2 n n in other words a is the sum of the kronecker products of u k with their complex conjugates we show the following sharp inequality as an application we show that the natural representation of u n n 1 acting on l 2 of the unit sphere in c n and restricted to mean zero functions satisfies for any choice 1 n in u n the lower bound this extends a result due to lubotzky phillips and sarnak who proved this with so 3 in the place of u n

c algebras of real rank zero abstract the concept of real rank of a c algebra is introduced as a non commutative analogue of dimension it is shown that real rank zero is equivalent to the previously defined conditions fs and hp and that it is invariant under strong morita equivalence in particular under stable isomorphism real rank zero is also invariant under inductive limits and split extensions and the class may well be regarded as the conceptual completion of the af algebras in some cases notably when the algebra is matroid it is shown that the multiplier algebra also has real rank zero although that is not true in general by a result of g j murphy this implies a weyl von neumann type result for self adjoint multiplier elements in these cases

eigenvectors and eigenvalues in a random subspace of a tensor product given two positive integers n and k and a parameter t 0 1 we choose at random a vector subspace v n k n of dimension n tnk we show that the set of k tuples of singular values of all unit vectors in v n fills asymptotically as n tends to infinity a deterministic convex set k k t that we describe using a new norm in k

on the structure of simple c algebras tensored with a uhf algebra ii let d be a simple unital c algebra let b be a uhf algebra and put a b d it is proved that if p q a are projections and p q for all quasitraces on a then p q in the sense of murray and von neumann a more general result involving positive operators in a is also proved if d has finitely many extremal quasi traces and the projections in d k separate these then it is proved that a has real rank zero finally it is proved that provided d is stably finite then each positive state on k0 d lifts to a quasi trace

semi invertible extensions of c algebras abstract we prolong the list of c algebras which have the property that all extensions by a stable c algebra are semi invertible in particular it is shown to include group c algebras both reduced and full of certain amalgamated free products of amenable groups as well as all free products of nuclear c algebras with amalgamation over a common nuclear c subalgebra

no eigenvalues outside the limiting support of generally correlated gaussian matrices this paper investigates the behaviour of the spectrum of generally correlated gaussian random matrices whose columns are zero mean independent vectors but have different correlations under the specific regime where the number of their columns and that of their rows grow at infinity with the same pace following the approach proposed by vallet et al we prove that under some mild conditions there is no eigenvalue outside the limiting support of generally correlated gaussian matrices as an outcome of this result we establish that the smallest singular value of these matrices is almost surely greater than zero from a practical perspective this control of the smallest singular value is paramount to applications from statistical signal processing and wireless communication in which this kind of matrices naturally arise

dimension and stable rank in the k theory of c algebras in topological k theory which can be viewed as the algebraic side of the theory of vector bundles some of the interesting properties which one investigates are for example the conditions under which bundles must possess trivial direct summands or the extent to which the cancellation property for direct whitney sums of bundles holds such properties turn out to be controlled in part by the dimension of the base space and results describing the nature of this control are among what are frequently called stability results 17 during the past few years there has been a sudden flowering of a theory for c algebras see surveys 35 36 37 since c algebras are profitably thought of as non commutative locally compact spaces with the finitely generated projective modules being the appropriate generalization of vector bundles according to swan s theorem 40 it would be natural to look for stability results for c algebras but until now the theory has been focussed on the c groups of c algebras and there has been little discussion of stability properties presumably in part for lack of an appropriate concept of dimension for c algebras one of the main objectives of this paper is to introduce a concept of dimension for c algebras which directly generalizes the classical concept of dimension for compact spaces 16 20 22 and to develop some techniques for calculating it notably for c algebras which are obtained as crossed product algebras for an action of the group of integers in algebraic k theory which is the generalization of topological c theory to rings a substantial number of stability properties have been discovered 2 39 42 48 within that theory the concept which has played a role most analogous to that of dimension is the bass stable rank but despite its many successes the bass stable rank has often been difficult to calculate even for relatively uncomplicated rings another main objective of the present paper is to make accessible for application to c algebras some of the stability results from algebraic c theory this comes about from the fact that the concept of dimension which we introduce dominates the bass stable rank i was led to seek stability results for c algebras by my desire to determine whether the cancellation property for projective modules holds for the irrational rotation c algebras discussed in 30 which as elliott 11 has suggested can appropriately be called 2 dimensional non commutative tori in more concrete terms the question is whether two projections in such an algebra which have the same trace must be unitarily equivalent the present paper provides the general theory which is needed for the affirmative answer which i obtained in a subsequent paper i will derive the

the free entropy dimension of hyperfinite von neumann algebras suppose m is a hyperfinite von neumann algebra with a normal tracial state and 1 n is a set of selfadjoint generators for m we calculate 0 1 n the modified free entropy dimension of 1 n moreover we show that 0 1 n depends only on m and consequently 0 1 n is independent of the choice of generators for m in the course of the argument we show that if b 1 b n is a set of selfadjoint generators for a von neumann algebra r with a normal tracial state and b 1 b n has finite dimensional approximants then 0 n 0 b 1 b n for any hyperfinite von neumann subalgebra n of r combined with a result by voiculescu this implies that if r has a regular diffuse hyperfinite von neumann subalgebra then 0 b 1 b n 1

generalized inductive limits of quasidiagonal c algebras if a is a unital quasidiagonal c algebra we construct a generalized inductive limit ba which is simple unital and inherits many structural properties from a if a is the unitization of a non simple purely infinite algebra e g the cone over a cuntz algebra then ba is tracially af which among other things lends support to a conjecture of toms

cauchy noise loss for stochastic optimization of random matrix models via free deterministic equivalents abstract for random matrix models the parameter estimation based on the traditional likelihood functions is not straightforward in particular when we have only one sample matrix we introduce a new parameter optimization method for random matrix models which works even in such a case the method is based on the spectral distribution instead of the traditional likelihood in the method the cauchy noise has an essential role because the free deterministic equivalent which is a tool in free probability theory allows us to approximate the spectral distribution perturbed by cauchy noises by a smooth and accessible density function moreover we study an asymptotic property of determination gap which has a similar role as generalization gap besides we propose a new dimensionality recovery method for the signal plus noise model and experimentally demonstrate that it recovers the rank of the signal part even if the true rank is not small it is a simultaneous rank selection and parameter estimation procedure

computing norms of free operators with matrix coefficients let g 1 g 2 g n be the generators of the free group f n and denote by its left regular representation a formula for norms of operators of the form a i g i where a i are complex matrices or operators is derived similarly to the formula of akemann and ostrand for the scalar case this formula can be used to compute norms of arbitrary finitely supported convolution operators on the free group essentially the same technique yields a formula for the norms of sums of matrix valued creation and annihilation operators on full fock space finally some results about the numerical evaluation of the formula are proved

mixed moments of voiculescu s gaussian random matrices abstract we prove an explicit combinatorial formula for the mixed moments of a family of independent selfadjoint gaussian random matrices as a corollary we show that the convergence of such mixed moments proved by voiculescu is as fast as o 1 n 2 and we use this fact to give a short proof of the fact recently proved by hiai and petz that the convergence in voiculescu s result actually holds almost surely this latter result is then used to give an alternative proof of the result by s wassermann that free group factors embed into ultra products of matrix algebras moreover the almost sure convergence result is used to obtain asymptotic lower respectively upper bounds on the maximum respectively minimum of the spectrum of s s for gaussian random matrices s with operator entries in particular those studied by u haagerup and the author 1999 documenta math 4 341 450

on non semisplit extensions tensor products and exactness of group c algebras we show the existence of a block diagonal extensionb of the suspensions a of the reduced groupc algebraa c sl 2 such that there is only onec norm on the algebraic tensor productb op b butb is not nuclear even not exact thus the class of exactc algebras is not closed under extensions the existence comes from a new established tensorial duality between the weak expectation property wep of lance and the local variant llp of the lifting property we characterize the local lifting property of separable unitalc algebrasa as follows a has the local lifting property if and only if ext s a is a group wheres a is the suspension ofa if moreovera is the quotient algebra of ac algebra withwep for brevity a isqwep but does not satisfyllp then there exists a quasidiagonal extensionb of the suspensions a by the compact operators such that on the algebraic tensor productb op b there is only onec norm the question if everyc algebra isqwep remains open but we obtain the following results onqwep ac algebrac isqwep if and only ifc isqwep a von neumannii 1 factorn with separable predualn isqwep if and only ifn is a von neumann subfactor of the ultrapower of the hyperfiniteii 1 factor ifg is a maximally almost periodic discrete non amenable group with haagerup s herz schur multiplier constant g 1 then the universal groupc algebrac g is not exact but the reduced groupc albegrac g is exact and isqwep but does not satisfywep andllp we study functiorial properties of the classes ofc algebras satisfyingwep llp resp beingqwep as applications we obtain some unexpected relations between some open questions onc algebras

the norm of polynomials in large random and deterministic matrices let bf x n x 1 n ldots x p n be a family of n n independent normalized random matrices from the gaussian unitary ensemble we state sufficient conditions on matrices bf y n y 1 n ldots y q n possibly random but independent of xn for which the operator norm of p bf x n bf y n bf y n converges almost surely for all polynomials p limits are described by operator norms of objects from free probability theory taking advantage of the choice of the matrices yn and of the polynomials p we get for a large class of matrices the no eigenvalues outside a neighborhood of the limiting spectrum phenomena we give examples of diagonal matrices yn for which the convergence holds convergence of the operator norm is shown to hold for block matrices even with rectangular gaussian blocks a situation including non white wishart matrices and some matrices encountered in mimo systems

multipliers of the fourier algebras of some simple lie groups and their discrete subgroups for any amenable locally compact group g the space of multipliers ma g of the fourier algebra a g coincides with the space b g of functions on g that are linear combinations of continuous positive definite functions we prove that ma g b g 0 for many non amenable connected groups more specifically we prove that moa g b g 0 for the classical complex lie groups and the general lorentz groups soo n 1 n 2 moa g is a certain subspace of ma g which we call the space of completely bounded multipliers of a g unlike ma g the space moa g has nice stability properties with respect to direct products of groups it is known that the fourier algebra of the free group on n generators n 2 2 admits an unbounded approximate unit pn which is bounded in the multiplier norm we extend this result to any closed subgroup of the general lorentz group soo n 1 moreover we show that for these groups pn can be chosen to be bounded with respect to the moa g norm by a duality argument we obtain that the reduced c algebra of every discrete subgroup of soo n 1 has the completely bounded approximation property in particular this property holds for c f2 the reduced c algebra of the free group on two generators we also prove

regularity properties in the classification program for separable amenable c algebras we report on recent progress in the program to classify separable amenable c algebras our emphasis is on the newly apparent role of regularity properties such as finite decomposition rank strict comparison of positive elements and z stability and on the importance of the cuntz semigroup we include a brief history of the program s successes since 1989 a more detailed look at the villadsen type algebras which have so dramatically changed the landscape and a collection of announcements on the structure and properties of the cuntz semigroup

a note on an inequality involving the normal distribution the following inequality is useful in studying a variation of the classical isoperimetric problem let x be normally distributed with mean 0 and variance 1 if g is absolutely continuous and g x has finite variance then e lbrack g x rbrack 2 geq operatorname var lbrack g x rbrack with equality if and only if g x is linear in x the proof involves expanding g x in hermite polynomials

lagrange mesh calculations in momentum space the lagrange mesh method is a powerful method to solve eigenequations written in configuration space it is very easy to implement and very accurate using a gauss quadrature rule the method requires only the evaluation of the potential at some mesh points the eigenfunctions are expanded in terms of regularized lagrange functions which vanish at all mesh points except one it is shown that this method can be adapted to solve eigenequations written in momentum space keeping the convenience and the accuracy of the original technique in particular the kinetic operator is a diagonal matrix observables and wave functions in both configuration space and momentum space can also be easily computed with good accuracy using only eigenfunctions computed in the momentum space the method is tested with gaussian and yukawa potentials requiring respectively a small and a large mesh to reach convergence corresponding wave functions in both spaces are compared with each other using the fourier transform

an lp version of von neumann dimension for banach space representations of sofic groups abstract in 5 a gournay defined a notion of l p dimension for invariant subspaces of l q n with amenable the number dim l q l p v is dim v when p q and is preserved by a certain class of equivariant bounded linear isomorphisms we develop a notion of dim l p y where y is a banach space with a uniformly bounded action of a sofic group and is a sofic approximation in particular our definition makes sense for a large class of non amenable groups we also develop a notion of dim s p y with an r embeddable group and s p the space of finite dimensional schatten p class operators these numbers are invariant under bounded equivariant linear isomorphisms and under the natural translation action of dim l p l p v dim v and dim s p l p v dim v for 1 p 2 in particular this shows that l p v is not isomorphic to l p w as a representation of if dim v dim w and is r embeddable we discuss other concrete computations in a follow up paper including proving that our dimension agrees with von neumann dimension for representations contained in a multiple of the left regular representation

tensor products of c algebras with the ideal property abstract combining a construction of dadarlat of a unital simple non exact c algebra c of real rank zero and stable rank one which is shape equivalent to a uhf algebra with results of kirchberg and a result obtained by dadarlat and the first named author we show that b h c contains an ideal that is not generated by its projections we also find a unital separable sub c algebra a of b h such that a is of real rank zero and a c has an ideal that is not generated by its projections

the spectral edge of unitary brownian motion the brownian motion u n t t ge 0 on the unitary group converges as a process to the free unitary brownian motion u t t ge 0 as n rightarrow infty in this paper we prove that it converges strongly as a process not only in distribution but also in operator norm in particular for a fixed time t 0 we prove that the unitary brownian motion has a spectral edge there are no outlier eigenvalues in the limit we also prove an extension theorem any strongly convergent collection of random matrix ensembles independent from a unitary brownian motion also converge strongly jointly with the brownian motion we give an application of this strong convergence to the jacobi process

topological free entropy dimension in unital c algebras abstract the notion of topological free entropy dimension of n tuple of elements in a unital c algebra was introduced by voiculescu in the paper we compute topological free entropy dimension of one self adjoint element and topological free orbit dimension of one self adjoint element in a unital c algebra we also calculate the values of topological free entropy dimensions of any families of self adjoint generators of some unital c algebras including irrational rotation c algebra uhf algebra and minimal tensor product of two reduced c algebras of free groups

a new proof of the equivalence of injectivity and hyperfiniteness for factors on a separable hilbert space abstract in 1975 a connes proved the fundamental result that injective factors on a separable hilbert space are hyperfinite in this paper a new proof of this result is presented in which the most technical parts of connes proof are avoided particularly the proof does not rely on automorphism group theory the starting point in this approach is wassermann s simple proof of injective semidiscrete together with choi and effros characterization of semidiscrete von neumann algebras as those von neumann algebras n for which the identity map on n has an approximate completely positive factorization through n n matrices

an approximate universal coefficient theorem an approximate universal coefficient theorem auct for certain c algebras is established we present a proof that kirchberg phillips s classification theorem for separable nuclear purely infinite simple c algebras is valid for c algebras satisfying the auct instead of the uct it is proved that two versions of auct are in fact the same we also show that c algebras that are locally approximated by c algebras satisfying the auct satisfy the auct as an application we prove that certain simple c algebras which are locally type i are in fact isomorphic to simple ah algebras as another application we show that a sequence of residually finite dimensional c algebras which are asymptotically nuclear and which asymptotically satisfies the auct can be embedded into the same simple af algebra

extensions of the reduced group c algebra of a free product of amenable groups abstract we prove that the unitary equivalence classes of extensions of c r g by any unital stable c algebra taken modulo extensions which split via an asymptotic homomorphism form a group which can be calculated from the universal coefficient theorem of kk theory when g is a free product of a countable collection of countable amenable groups

on the classification of nuclear c algebras two of the most influential works on c algebras from the mid seventies brown douglas and fillmore s 6 and elliott s 21 both contain uniqueness and existence results in the now standard sense which we shall outline below these papers served as keystones for two separate theories kk theory and the classification program which for many years parted ways with only moderate interaction but with this common origin in mind it is not surprising that recent years have seen a fruitful interaction which has been one of the main engines behind rapid progress in the classification program in the present paper we take this interaction even further combining a concept of quasidiagonality for representations and a new characterization of equivalence in the cuntz picture of kk theory we achieve a general uniqueness result and from the quasidiagonality notion we derive a corresponding general existence result by comparing well known realizations of the kk groups these results are then employed to obtain new classification results for certain classes of quasidiagonal c algebras introduced by h lin an important novel feature of these classes is that they are defined by a certain local approximation property rather than by an inductive limit construction we emphasize that our fundamental uniqueness result does not depend on the universal coefficient theorem uct nor on nuclearity of the c algebras involved on the other hand when we refine the uniqueness result for use in classification we need to gradually add such conditions on the c algebras it is to be expected that requirements of this type are necessary but we have found that holding them back as long as possible leads to more conceptual proofs while clarifying the role of the uct and nuclearity in classification further since uniqueness results answer fundamental questions about kk and k theory they are of interest in themselves we have pursued this theme in 14 and defer to this paper the proof of the new characterization of kk theory which lies behind our uniqueness results our existence and uniqueness results are in the spirit of the classical ext theory from 6 the main complication overcome in the paper is to control the stabilization which is necessary when one works with finite c algebras in the infinite case where programs of this type have already been successfully carried out stabilization is unnecessary yet our methods are sufficiently versatile to allow us to reprove from a handful of basic results the classification of purely infinite nuclear c algebras of kirchberg and phillips

the single ring theorem we study the empirical measure lan of the eigenvalues of non normal square matrices of the form an untnvn with un vn independent haar distributed on the unitary group and tn real diagonal we show that when the empirical measure of the eigenvalues of tn converges and tn satisfies some technical conditions lan converges towards a rotationally invariant measure on the complex plane whose support is a single ring in particular we provide a complete proof of feinberg zee single ring theorem 6 we also consider the case where un vn are independent haar distributed on the orthogonal group

topological dynamical systems associated to ii1 factors abstract if n r is a separable ii1 factor the space h om n r of unitary equivalence classes of unital homomorphisms n r is shown to have a surprisingly rich structure if n is not hyperfinite h om n r is an infinite dimensional complete metrizable topological space with convex like structure and the outer automorphism group out n acts on it by affine homeomorphisms if n r then h om n r is just a point property t is reflected in the extreme points they re discrete in this case for certain free products n r every countable group acts nontrivially on h om n r and we show the extreme points are not discrete for these examples finally we prove that the dynamical systems associated to free group factors are isomorphic