crossed products of nuclear c algebras and their traces abstract this work addresses the blackadar kirchberg question bkq as it pertains to group and reduced crossed product c algebras the bkq asks if every stably finite c algebra admits the matricial field mf property we characterize the mf property for a class of reduced crossed product c algebras and for their traces we prove that every ordered abelian group is in a sense mf and show that there is no k theoretic obstruction to an affirmative answer to the bkq using classification techniques and induced k theoretic dynamics we give an affirmative answer to the bkq for reduced crossed products of ah algebras of real rank zero by free groups combining our results with recent progress in elliott s classification program shows that the reduced crossed product of a separable simple unital nuclear uct and monotracial c algebra by the free group is always mf we also examine traces on these crossed products and show they always admit certain finite dimensional approximation properties by appealing to a result of ozawa rordam and sato we show that semidirect products of discrete amenable groups by free groups admit mf reduced group c algebras

the h vectors of arithmetically gorenstein sets of points on a general sextic surface in p3 abstract we give the possible h vectors of arithmetically gorenstein sets of points on a general sextic surface in p 3 and show the existence of such a set for each possible h vector our methods are centered on the vector bundle techniques developed by several people notably chiantini and faenzi together with extensive use of liaison theory

cayley bacharach schemes and their canonical modules a set of s points in p d is called a cayley bacharach scheme cb scheme if every subset of s 1 points has the same hilbert function we investigate the consequences of this weak uniformity the main result characterizes cb schemes in terms of the structure of the canonical module of their projective coordinate ring from this we get that the hilbert function of a cb scheme x has to satisfy growth conditions which are only slightly weaker than the ones given by harris and eisenbud for points with the uniform position property we also characterize cb schemes in terms of the conductor of the projective coordinate ring in its integral closure and in terms of the forms of minimal degree passing through a linked set of points

a proximal point algorithm for log determinant optimization with group lasso regularization we consider the covariance selection problem where variables are clustered into groups and the inverse covariance matrix is expected to have a blockwise sparse structure this problem is realized via penalizing the maximum likelihood estimation of the inverse covariance ma trix by group lasso regularization we propose to solve the resulting log determinant optimization problem with the classical proximal point algorithm ppa at each iteration as it is difficult to update the primal variables directly we first solve the dual subproblem by an inexact semismooth newton cg method and then update the primal variables by explicit formulas based on the com puted dual variables we also propose to accelerate the ppa by an inexact generalized newton s method when the iterate is close to the solution theoretically we prove that at the optimal solution the nonsingularity of the generalized hessian matrices of the dual subproblem is equivalent to the constraint nondegeneracy condition for the primal problem global and local convergence results are also presented for the proposed ppa moreover based on the augmented lagrangian function of the dual problem we derive an alternating direction method adm which is easily implementable and is demonstrated to be efficient for random problems numerical results including comparisons with the adm on both synthetic and real data are presented to demonstrate that the proposed newton cg based ppa is stable and efficient and in particular outperforms the adm when high accuracy is required

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

universality in several matrix models via approximate transport maps we construct approximate transport maps for perturbative several matrix models as a consequence we deduce that local statistics have the same asymptotic as in the case of independent gue or goe matrices i e they are given by the sine kernel in the bulk and the tracy widom distribution at the edge and we show averaged energy universality i e universality for averages of m points correlation functions around some energy level e in the bulk as a corollary these results yield universality for self adjoint polynomials in several independent gue or goe matrices which are close to the identity

semi infinite programming theory methods and applications starting from a number of motivating and abundant applications in 2 including control of robots eigenvalue computations mechanical stress of materials and statistical design the authors describe a class of optimization problems which are referred to as semi infinite because their constraints bound functions of a finite number of variables on a whole region in 3 5 first and second order optimality conditions are derived for general nonlinear problems as well as a procedure for reducing the problem locally to one with only finitely many constraints another main effort for achieving simplification is through duality in 6 there algebraic properties of finite linear programming are brought to bear on duality theory in semi infinite programming section 7 treats numerical methods based on either discretization or local reduction with the emphasis on the design of superlinearly convergent sqp type methods taking this differentiable point of view this paper can be considered to be complementar

mf actions and k theoretic dynamics we study the interplay of c dynamics and k theory notions of chain recurrence for transformations groups x and mf actions for non commutative c dynamical systems a are translated into k theoretical language where purely algebraic conditions are shown to be necessary and sufficient for a reduced crossed product to admit norm microstates we are particularly interested in actions of free groups on af algebras in which case we prove that a k theoretic coboundary condition determines whether or not the reduced crossed product is a matricial field mf algebra one upshot is the equivalence of stable finiteness and being mf for these reduced crossed product algebras

grothendieck s theorem past and present probably the most famous of grothendieck s contributions to banach space theory is the result that he himself described as the fundamental theorem in the metric theory of tensor products that is now commonly referred to as grothendieck s theorem gt in short or sometimes as grothendieck s inequality this had a major impact first in banach space theory roughly after 1968 then later on in c algebra theory roughly after 1978 more recently in this millennium a new version of gt has been successfully developed in the framework of operator spaces or non commutative banach spaces in addition gt independently surfaced in several quite unrelated fields in connection with bell s inequality in quantum mechanics in graph theory where the grothendieck constant of a graph has been introduced and in computer science where the grothendieck inequality is invoked to replace certain np hard problems by others that can be treated by semidefinite programming and hence solved in polynomial time in this expository paper we present a review of all these topics starting from the original gt we concentrate on the more recent developments and merely outline those of the first banach space period since detailed accounts of that are already available for instance the author s 1986 cbms notes

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

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

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

rank 2 arithmetically cohen macaulay bundles on a nonsingular cubic surface abstract rank 2 indecomposable arithmetically cohen macaulay bundles e on a nonsingular cubic surface x in p 3 are classified by means of the possible forms taken by the minimal graded free resolution of e over p 3 the admissible values of the chern classes of e are listed and the vanishing locus of a general section of e is studied properties of e such as slope semi stability and simplicity are investigated the number of relevant families is computed together with their dimension

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

discretization in semi infinite programming the rate of convergence the discretization approach for solving semi infinite optimization problems is considered we are interested in the convergence rate of the error between the solution of the semi infinite problem and the solution of the discretized program depending on the discretization mesh size it will be shown how this rate depends on whether the minimizer is strict of order one or two and on whether the discretization includes boundary points of the index set in a specific way this is done for ordinary and for generalized semi infinite problems

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 the global fluctuations of block gaussian matrices in this paper we study the global fluctuations of block gaussian matrices within the framework of second order free probability theory in order to compute the second order cauchy transform of these matrices we introduce a matricial second order conditional expectation and compute the matricial second order cauchy transform of a certain type of non commutative random variables as a by product using the linearization technique we obtain the second order cauchy transform of non commutative rational functions evaluated on selfadjoint gaussian matrices

non commutative polynomials of independent gaussian random matrices the real and symplectic cases in ht2 haagerup and thorbjo rnsen prove the following extension of voiculescu s random matrix model cf v2 theorem 2 2 for each n let x1 n xr n be r independent complex self adjoint random matrices from the class open image in new window and let x1 xr be a semicircular system in a c probability space then for any polynomial p in r non commuting variables the convergence

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

operator valued chordal loewner chains and non commutative probability abstract we adapt the theory of chordal loewner chains to the operator valued matricial upper half plane over a c algebra a we define an a valued chordal loewner chain as a subordination chain of analytic self maps of the a valued upper half plane such that each f t is the reciprocal cauchy transform of an a valued law t such that the mean and variance of t are continuous functions of t we relate a valued loewner chains to processes with a valued free or monotone independent increments just as was done in the scalar case by bauer 8 and schleisinger 48 we show that the loewner equation t f t z d f t z v t z when interpreted in a certain distributional sense defines a bijection between lipschitz mean zero loewner chains f t and vector fields v t z of the form v t z g t z where t is a generalized a valued law based on the loewner equation we derive a combinatorial expression for the moments of t in terms of t we also construct non commutative random variables on an operator valued monotone fock space which realize the laws t finally we prove a version of the monotone central limit theorem which describes the behavior of f t as t when t has uniformly bounded support

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

a globally convergent sqp method for semi infinite nonlinear optimization abstract a new approach for semi infinite programming problems is presented which belongs to the class of successive quadratic programming sqp methods with trust region technique the proposed algorithm employs the exact l penalty function as a criterion function and incorporates an appropriate scheme for estimating active constraints it is proved that the algorithm is globally convergent under some assumptions numerical experiments show that the algorithm is very promising in practice

second order freeness and fluctuations of random matrices i gaussian and wishart matrices and cyclic fock spaces abstract we extend the relation between random matrices and free probability theory from the level of expectations to the level of fluctuations we introduce the concept of second order freeness and interpret the global fluctuations of gaussian and wishart random matrices by a general limit theorem for second order freeness by introducing cyclic fock space we also give an operator algebraic model for the fluctuations of our random matrices in terms of the usual creation annihilation and preservation operators we show that orthogonal families of gaussian and wishart random matrices are asymptotically free of second order

a regularized explicit exchange method for semi infinite programs with an infinite number of conic constraints the semi infinite program sip is normally represented with infinitely many inequality constraints and has been studied extensively so far however there have been very few studies on the sip involving conic constraints even though it has important applications such as chebyshev like approximation filter design and so on in this paper we focus on the sip with infinitely many conic constraints called an sicp for short we show that under the robinson constraint qualification a local optimum of the sicp satisfies the kkt conditions that can be represented only with a finite subset of the conic constraints we also introduce two exchange type algorithms for solving the convex sicp we first provide an explicit exchange method and show that it has global convergence under the strict convexity assumption on the objective function we then propose an algorithm combining a regularization method with the explicit exchange method and establish global convergence of the hybrid algorithm without the strict c

hilbert function of powers of ideals of low codimension we study the hilbert function of the powers of homogeneous ideals which are either cohen macaulay of codimension 2 or gorenstein of codimension 3 we show that that if i is an ideal in one of these classes and it is of linear type then for all k the hilbert function of i k depends only on the hilbert function of i in other words if i andj are ideals in one of the above mentioned classes which are both of linear type and they have the same hilbert function then also i k and j k have the same hilbert function for all k

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

hilbert functions of irreducible arithmetically gorenstein schemes abstract in this paper we compute the hilbert functions of irreducible or smooth and reduced arithmetically gorenstein schemes that are twisted anti canonical divisors on arithmetically cohen macaulay schemes we also prove some folklore results characterizing the hilbert functions of irreducible standard determinantal schemes and we use them to produce a new class of functions that occur as hilbert functions of irreducible or smooth and reduced arithmetically gorenstein schemes in any codimension

on the convergence of the extremal eigenvalues of empirical covariance matrices with dependence consider a sample of a centered random vector with unit covariance matrix we show that under certain regularity assumptions and up to a natural scaling the smallest and the largest eigenvalues of the empirical covariance matrix converge when the dimension and the sample size both tend to infinity to the left and right edges of the marchenko pastur distribution the assumptions are related to tails of norms of orthogonal projections they cover isotropic log concave random vectors as well as random vectors with i i d coordinates with almost optimal moment conditions the method is a refinement of the rank one update approach used by srivastava and vershynin to produce non asymptotic quantitative estimates in other words we provide a new proof of the bai and yin theorem using basic tools from probability theory and linear algebra together with a new extension of this theorem to random matrices with dependent entries

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

random quantum channels ii entanglement of random subspaces renyi entropy estimates and additivity problems in this paper we obtain new bounds for the minimum output entropies of random quantum channels these bounds rely on random matrix techniques arising from free probability theory we then revisit the counterexamples developed by hayden and winter to get violations of the additivity equalities for minimum output renyi entropies we show that random channels obtained by randomly coupling the input to a qubit violate the additivity of the p renyi entropy for some sequences of random quantum channels we compute almost surely the limit of their schatten s 1 to s p norms

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

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

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

secant varieties of the varieties of reducible hypersurfaces in pn given the space v p d n 1 n 1 1 of forms of degree d in n variables and given an integer l 1 and a partition of d d1 dr it is in general an open problem to obtain the dimensions of the l secant varieties l xn 1 for the subvariety xn 1 v of hypersurfaces whose defining forms have a factorization into forms of degrees d1 dr modifying a method from intersection theory we relate this problem to the study of the weak lefschetz property for a class of graded algebras based on which we give a conjectural formula for the dimension of l xn 1 for any choice of parameters n l and this conjecture gives a unifying framework subsuming all known results moreover we unconditionally prove the formula in many cases considerably extending previous results as a consequence of which we verify many special cases of previously posed conjectures for dimensions of secant varieties of segre varieties in the special case of a partition with two parts i e r 2 we also relate this problem to a conjecture by froberg on the hilbert function of an ideal generated by general forms

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

inverse systems of zero dimensional schemes in p n abstract the authors construct the global macaulay inverse system l z for a zero dimensional subscheme z of projective n space p n over an algebraically closed field k from the local inverse systems of the irreducible components of z they show that when z is locally gorenstein a generic element f of degree d apolar to z determines z if d is larger than an invariant z as a consequence of this globalization they show that a natural upper bound for the hilbert function of gorenstein artin quotients of the coordinate ring of z is achieved for large socle degree they also show the uniqueness of generalized additive decompositions of a homogeneous form into powers of linear forms under suitable hypotheses the main tools are elementary but delicate they involve a careful study of how to homogenize a local inverse system and of the behavior of the homogenization under a change of coordinates

an interior point sequential quadratic programming type method for log determinant semi infinite programs abstract in this paper we consider a nonlinear semi infinite program that minimizes a function including a log determinant logdet function over positive definite matrix constraints and infinitely many convex inequality constraints we call this problem siplog where sip stands for semi infinite program and log comes from log det function the main purpose of the paper is to develop an algorithm for efficiently computing a karush kuhn tucker kkt point for the siplog more specifically we propose an interior point sequential quadratic programming type method that solves inexactly a sequence of semi infinite quadratic programs approximating the siplog furthermore to generate a search direction in the dual matrix space associated with the semi definite constraint we solve scaled newton equations that yield the family of monteiro zhang directions we prove that the proposed method weakly converges to a kkt point under some mild assumptions finally we conduct some numerical experiments to demonstrate the efficiency of the proposed method

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

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

the projective geometry of the gale transform abstract the gale transform an involution on sets of points in projective space appears in a multitude of guises and in subjects as diverse as optimization coding theory theta functions and recently in our proof that certain general sets of points fail to satisfy the minimal free resolution conjecture see eisenbud and popescu 1999 invent math 136 419 449 in this paper we reexamine the gale transform in the light of modern algebraic geometry we give a more general definition in the context of finite locally gorenstein subschemes we put in modern form a number of the more remarkable examples discovered in the past and we add new constructions and connections to other areas of algebraic geometry we generalize goppa s theorem in coding theory and we give new applications to castelnuovo theory we also give references to classical and modern sources

eigenvalues of block structured asymmetric random matrices we study the spectrum of an asymmetric random matrix with block structured variances the rows and columns of the random square matrix are divided into d partitions with arbitrary size linear in n the parameters of the model are the variances of elements in each block summarized in g r d d using the hermitization approach and by studying the matrix valued stieltjes transform we show that these matrices have a circularly symmetric spectrum we give an explicit formula for their spectral radius and a set of implicit equations for the full density function we discuss applications of this model to neural networks

spectral properties of polynomials in independent wigner and deterministic matrices abstract on the one hand we prove that almost surely for large dimension there is no eigenvalue of a hermitian polynomial in independent wigner and deterministic matrices in any interval lying at some distance from the supports of a sequence of deterministic probability measures which is computed with the tools of free probability on the other hand we establish the strong asymptotic freeness of independent wigner matrices and any family of deterministic matrices with strong limiting distribution

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

all extensions of c _r left mathbb f_n right are semi invertible it is shown that an extension of the reduced group c algebra of a free group by the compact operators can be made asymptotically split by addition of another extension which admits a completely positive lifting

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

local convergence of sqp methods in semi infinite programming in this paper we begin with pointing out how a semi infinite programming problem can be reduced locally to a problem of finite dimensional programming such a reduction has the advantage that efficient numerical methods like sequential quadratic programming sqp methods can be applied however the reduced problem involves constraint functions that are defined only implicitly values of these functions and their derivatives must be computed iteratively with controllable errors we interpret them as perturbations of the correct constraints and apply an sqp method with a broyden fletcher goldfarb shanno bfgs update extending the convergence analysis by fontecilla steihaug and tapia for these methods to include perturbations of the constraints and their derivatives we are able to show q superlinear convergence and at the same time to indicate at which rate the error in the calculation of the constraints must be reduced as the iteration progresses

local laws for polynomials of wigner matrices abstract we consider general self adjoint polynomials in several independent random matrices whose entries are centered and have the same variance we show that under certain conditions the local law holds up to the optimal scale i e the eigenvalue density on scales just above the eigenvalue spacing follows the global density of states which is determined by free probability theory we prove that these conditions hold for general homogeneous polynomials of degree two and for symmetrized products of independent matrices with i i d entries thus establishing the optimal bulk local law for these classes of ensembles in particular we generalize a similar result of anderson for anticommutator for more general polynomials our conditions are effectively checkable numerically

arithmetically cohen macaulay bundles and sets of points on general surfaces abstract the first chern class of an initialized rank 2 bundle determines its level of stability we determine bounds for numbers c 1 0 such that on a general surface x p 3 of degree d 4 there exists a rank 2 indecomposable stable acm bundle e on x with first chern class c 1 e c 1

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

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

improved subspace estimation for multivariate observations of high dimension the deterministic signals case we consider the problem of subspace estimation in situations where the number of available snapshots and the observation dimension are comparable in magnitude in this context traditional subspace methods tend to fail because the eigenvectors of the sample correlation matrix are heavily biased with respect to the true ones it has recently been suggested that this situation where the sample size is small compared to the observation dimension can be very accurately modeled by considering the asymptotic regime where the observation dimension m and the number of snapshots n converge to at the same rate using large random matrix theory results it can be shown that traditional subspace estimates are not consistent in this asymptotic regime furthermore new consistent subspace estimate can be proposed which outperform the standard subspace methods for realistic values of m and n the work carried out so far in this area has always been based on the assumption that the observations are random independent and identically distributed in the time domain the goal of this paper is to propose new consistent subspace estimators for the case where the source signals are modelled as unknown deterministic signals in practice this allows to use the proposed approach regardless of the statistical properties of the source signals in order to construct the proposed estimators new technical results concerning the almost sure location of the eigenvalues of sample covariance matrices of information plus noise complex gaussian models are established these results are believed to be of independent interest

holder continuity of cumulative distribution functions for noncommutative polynomials under finite free fisher information abstract this paper contributes to the current studies on regularity properties of noncommutative distributions in free probability theory more precisely we consider evaluations of selfadjoint noncommutative polynomials in noncommutative random variables that have finite non microstates free fisher information highlighting the special case of lipschitz conjugate variables for the first time in this generality it is shown that the analytic distributions of those evaluations have holder continuous cumulative distribution functions with an explicit holder exponent that depends only on the degree of the considered polynomial for linear polynomials we reach in the case of finite non microstates free fisher information the optimal holder exponent 2 3 and get lipschitz continuity in the case of lipschitz conjugate variables in particular our results guarantee that such polynomial evaluations have finite logarithmic energy and thus finite non microstates free entropy which partially settles a conjecture of charlesworth and shlyakhtenko 8 we further provide a very general criterion that gives for weak approximations of measures having holder continuous cumulative distribution functions explicit rates of convergence in terms of the kolmogorov distance finally we combine these results to study the asymptotic eigenvalue distributions of polynomials in gues or matrices with more general gibbs laws for gibbs laws this extends the corresponding result obtained in 21 from convergence in distribution to convergence in kolmogorov distance in the gue case we even provide explicit rates which quantify results of 23 24 in terms of the kolmogorov distance

a random matrix approach to the lack of projections in cred f2 abstract in 1982 pimsner and voiculescu computed the k 0 and k 1 groups of the reduced group c algebra c red f k of the free group f k on k generators and settled thereby a long standing conjecture c red f k has no projections except for the trivial projections 0 and 1 later simpler proofs of this conjecture were found by methods from k theory or from non commutative differential geometry in this paper we provide a new proof of the fact that c red f k is projectionless the new proof is based on random matrices and is obtained by a refinement of the methods recently used by the first and the third named author to show that the semigroup ext c red f k is not a group for k 2 by the same type of methods we also obtain that two phenomena proved by bai and silverstein for certain classes of random matrices no eigenvalues outside a small neighbourhood of the support of the limiting distribution and exact separation of eigenvalues by gaps in the limiting distribution also hold for arbitrary non commutative selfadjoint polynomials of independent gue goe or gse random matrices with matrix coefficients

applications of realizations aka linearizations to free probability abstract we show how the combination of new linearization ideas in free probability theory with the powerful realization machinery developed over the last 50 years in fields including systems engineering and automata theory allows solving the problem of determining the eigenvalue distribution or even the brown measure in the non selfadjoint case of noncommutative rational functions of random matrices when their size tends to infinity along the way we extend evaluations of noncommutative rational expressions from matrices to stably finite algebras e g type ii 1 von neumann algebras with a precise control of the domains of the rational expressions the paper provides sufficient background information with the intention that it should be accessible both to functional analysts and to algebraists

support convergence in the single ring theorem we study 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 all these eigenvalues lie in a single ring

eigenvalues of non hermitian random matrices and brown measure of non normal operators hermitian reduction and linearization method abstract we study the brown measure of certain non hermitian operators arising from voiculescu s free probability theory usually those operators appear as the limit in moments of certain ensembles of non hermitian random matrices and the brown measure gives then a canonical candidate for the limit eigenvalue distribution of the random matrices a prominent class for our operators is given by polynomials in free variables other explicit examples include r diagonal elements and elliptic elements for which the brown measure was already known and a new class of triangular elliptic elements our method for the calculation of the brown measure is based on a rigorous mathematical treatment of the hermitian reduction method as considered in the physical literature combined with subordination and the linearization trick

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