geometry of free loci and factorization of noncommutative polynomials abstract the free singularity locus of a noncommutative polynomial f is defined to be the sequence of hypersurfaces z n f x m n k g det f x 0 the main theorem of this article shows that f is irreducible if and only if z n f is eventually irreducible a key step in the proof is an irreducibility result for linear pencils arising from this is a free singularity locus nullstellensatz for noncommutative polynomials apart from consequences to factorization in a free algebra the paper also discusses its applications to invariant subspaces in perturbation theory and linear matrix inequalities in real algebraic geometry

convex matrix inequalities versus linear matrix inequalities most linear control problems lead directly to matrix inequalities mis many of these are badly behaved but a classical core of problems are expressible as linear matrix inequalities lmis in many engineering systems problems convexity has all of the advantages of a lmi since lmis have a structure which is seemingly much more rigid than convex mis there is the hope that a convexity based theory will be less restrictive than lmis how much more restrictive are lmis than convex mis there are two fundamentally different classes of linear systems problems dimension free and dimension dependent a dimension free mi is a mi where the unknowns are g tuples of matrices and appear in the formulas in a manner which respects matrix multiplication most of the classic mis of control theory are dimension free dimension dependent mis have unknowns which are tuples of numbers the two classes behave very differently and this survey describes what is known in each case about the relation between convex mis and lmis the proof techniques involve and necessitate new developments in the field of semialgebraic geometry

circular free spectrahedra abstract this paper considers matrix convex sets invariant under several types of rotations it is known that matrix convex sets that are free semialgebraic are solution sets of linear matrix inequalities lmis they are called free spectrahedra we classify all free spectrahedra that are circular that is closed under multiplication by e i t up to unitary equivalence the coefficients of a minimal lmi defining a circular free spectrahedron have a common block decomposition in which the only nonzero blocks are on the superdiagonal a matrix convex set is called free circular if it is closed under left multiplication by unitary matrices as a consequence of a hahn banach separation theorem for free circular matrix convex sets we show the coefficients of a minimal lmi defining a free circular free spectrahedron have up to unitary equivalence a block decomposition as above with only two blocks this paper also gives a classification of those noncommutative polynomials invariant under conjugating each coordinate by a different unitary matrix up to unitary equivalence such a polynomial must be a direct sum of univariate polynomials

on c algebras generated by isometries with twisted commutation relations in the theory of c algebras interesting noncommutative structures arise as deformations of the tensor product e g the rotation algebra a as a deformation of c s1 c s1 we deform the tensor product of two toeplitz algebras in the same way and study the universal c algebra t t generated by two isometries u and v such that uv e2 i vu and u v e 2 i vu for r since the second relation implies the first one we also consider the universal c algebra t t generated by two isometries u and v with the weaker relation uv e2 i vu such a weaker case does not exist in the case of unitaries and it turns out to be much more interesting than the twisted tensor product case t t we show that t t is nuclear whereas t t is not even exact also we compute the k groups and we obtain k0 t t z and k1 t t 0 and the same k groups for t t

some results related with statistical convergence and berezin symbols abstract the concept of discrete statistical abel convergence is introduced in terms of berezin symbols we present necessary and sufficient condition under which a series n 0 a n with bounded sequence a n n 0 of complex numbers is discrete statistically abel convergent by using concept of statistical convergence we also give slight strengthening of a result of gokhberg and krein on compact operators

non negative hereditary polynomials in a free algebra we prove a non negative stellensatz and a null stellensatz for a class of polynomials called hereditary polynomials in a free algebra

strong majorization in a free algebra we study in the spirit of modern real algebra the interplay between left ideals of the free algebra mathbb f with n generators and their suitably defined zero sets and similarly between quadratic submodules of mathbb f and their positivity sets

free holomorphic functions and interpolation in this paper we obtain a noncommutative multivariable analogue of the classical nevanlinna pick interpolation problem for analytic functions with positive real parts on the open unit disc given a function f lambda to mathbb c where lambda is an arbitrary subset of the open unit ball mathbb b n z in mathbb c n z 1 we find necessary and sufficient conditions for the existence of a free holomorphic function g with complex coefficients on the noncommutative open unit ball b mathcal h n 1 such that rm re g geq 0 quad rm and quad g z f z quad z in lambda where b mathcal h is the algebra of all bounded linear operators on a hilbert space mathcal h the proof employs several results from noncommutative multivariable operator theory and a noncommutative cayley transform introduced and studied in the present paper acting from the set of all free holomorphic functions with positive real parts to the set of all bounded free holomorphic functions all the results of this paper are obtained in the more general setting of free holomorphic functions with operator valued coefficients as consequences we deduce some results concerning operator valued analytic interpolation on the unit ball mathbb b n

free pluriharmonic majorants and commutant lifting in this paper we initiate the study of sub pluriharmonic curves and free pluriharmonic majorants on the noncommutative open ball b h n 1 x1 xn b h n x1x1 xnxn 1 2 1 where b h is the algebra of all bounded linear operators on a hilbert space h several classical results from complex analysis have analogues in this noncommutative multivariable setting we present basic properties for sub pluriharmonic curves characterize the class of sub pluriharmonic curves that admit free pluriharmonic majorants and find in this case the least free pluriharmonic majorants we show that for any free holomorphic function on b h n 1 the map 0 1 c r1 rn r rr1 rrn rr1 rrn is a sub pluriharmonic curve in the cuntz toeplitz algebra generated by the right creation operators r1 rn on the full fock space with n generators we prove that is in the noncommutative hardy space hball2 if and only if has a free pluriharmonic majorant in this case we find herglotz riesz and poisson type representations for the least pluriharmonic majorant of moreover we obtain a characterization of the unit ball of hball2 and provide a parametrization and concrete representations for all free pluriharmonic majorants of when is in the unit ball of hball2 in the second part of this paper we introduce a generalized noncommutative commutant lifting gncl problem which extends to our noncommutative multivariable setting several lifting problems including the classical sz nagy foias commutant lifting problem and the extensions obtained by treil volberg foias frazho kaashoek and biswas foias frazho as well as their multivariable noncommutative versions we solve the gncl problem and using the results regarding sub pluriharmonic curves and free pluriharmonic majorants on noncommutative balls we provide a complete description of all solutions in particular we obtain a concrete schur type description of all solutions in the noncommutative commutant lifting theorem

ideal convergence characterization of the completion of linear n normed spaces in this paper we give the completion of a linear n normed space in terms of ideal convergence by introducing the concept of a uniformly continuous n norm

on fixed points of self maps of the free ball abstract in this paper we study the structure of the fixed point sets of noncommutative self maps of the free ball we show that for such a map that fixes the origin the fixed point set on every level is the intersection of the ball with a linear subspace we provide an application for the completely isometric isomorphism problem of multiplier algebras of noncommutative complete pick spaces

the noncommutative lowner theorem for matrix monotone functions over operator systems abstract given a function f a b r lowner s theorem states f is monotone when extended to self adjoint matrices via the functional calculus if and only if f extends to a self map of the complex upper half plane in recent years several generalizations of lowner s theorem have been proven in several variables we use the relaxed agler m c carthy and young theorem on locally matrix monotone functions in several commuting variables to generalize results in the noncommutative case specifically we show that a real free function defined over an operator system must analytically continue to a noncommutative upper half plane as map into another noncommutative upper half plane

containment problems for polytopes and spectrahedra we study the computational question of whether a given polytope or spectrahedron s a as given by the positive semidefiniteness region of a linear matrix pencil a x is contained in another s b first we classify the computational complexity extending results on the polytope polytope case by gritzmann and klee to the polytope spectrahedron case for various restricted containment problems np hardness is shown we then study in detail semidefinite conditions to certify containment building upon work by ben tal and nemirovski and helton klep and mccullough in particular we discuss variations of a sufficient semidefinite condition to certify containment of a spectrahedron in a spectrahedron it is shown that these sufficient conditions even provide exact semidefinite characterizations for containment in several important cases including containment of a spectrahedron in a polyhedron moreover in the case of bounded s a the criteria will always succeed in certifying containment of some scale

nuclearity of semigroup c algebras and the connection to amenability abstract we study c algebras associated with subsemigroups of groups for a large class of such semigroups including positive cones in quasi lattice ordered groups and left ore semigroups we describe the corresponding semigroup c algebras as c algebras of inverse semigroups groupoid c algebras and full corners in associated group crossed products these descriptions allow us to characterize nuclearity of semigroup c algebras in terms of faithfulness of left regular representations and amenability of group actions moreover we also determine when boundary quotients of semigroup c algebras are uct kirchberg algebras this leads to a unified approach to cuntz algebras and ring c algebras

toeplitz algebras associated to endomorphisms of ore semigroups in this paper we consider the toeplitz algebra associated to actions of ore semigroups on c algebras in particular we consider injective and surjective actions of such semigroups we use the theory of groupoid dynamical systems to represent the toeplitz algebra as a groupoid crossed product we also discuss the k theory of the toeplitz algebra in some examples for instance we show that for the semigroup of positive matrices the k theory of the associated toeplitz algebra vanishes

c envelopes of semicrossed products by lattice ordered abelian semigroups abstract a semicrossed product is a non selfadjoint operator algebra encoding the action of a semigroup on an operator or c algebra we prove that when the positive cone of a discrete lattice ordered abelian group acts on a c algebra the c envelope of the associated semicrossed product is a full corner of a crossed product by the whole group by constructing a c cover that itself is a full corner of a crossed product and computing the shilov ideal we obtain an explicit description of the c envelope this generalizes a result of davidson fuller and kakariadis from z n to the class of all discrete lattice ordered abelian groups

the complex moment problem and subnormality a polar decomposition approach abstract it has been known that positive definiteness does not guarantee a bisequence to be a complex moment however it turns out that positive definite extendibility does theorems 1 and 22 and this is the main theme of this paper the main tool is generally understood polar decomposition to strengthen applicability of our approach we work out a criterion for positive definite extendibility in a fairly wide context theorems 9 and 29 all this enables us to prove characterizations of subnormality of unbounded operators having invariant domain theorems 37 and 39 and their further applications theorems 41 and 43 and a description of the complex moment problem on real algebraic curves theorems 52 and 56 the latter question is completed in the appendix in which we relate the complex moment problem to the two dimensional real one with emphasis on real algebraic sets

a positivstellensatz for non commutative polynomials a non commutative polynomial which is positive on a bounded semi algebraic set of operators has a weighted sum of squares representation this positivstellensatz parallels similar results in the commutative case a broader issue is to what extent does real semi algebraic geometry extend to non commutative polynomials our strict positivstellensatz is positive news on the opposite extreme from strict positivity would be a real nullstellensatz we give an example which shows that there is no non commutative real nullstellensatz along certain lines however we include a successful type of non commutative nullstellensatz proved by george bergman

ring c algebras we associate reduced and full c algebras to arbitrary rings and study the inner structure of these ring c algebras as a result we obtain conditions for them to be purely infinite and simple we also discuss several examples originally our motivation comes from algebraic number theory

limitations of semidefinite programs for separable states and entangled games semidefinite programs sdps are a framework for exact or approximate optimization with widespread application in quantum information theory we introduce a new method for using reductions to construct integrality gaps for sdps meaning instances where the sdp value is far from the true optimum these are based on new limitations on the sum of squares sos hierarchy in approximating two particularly important sets in quantum information theory where previously no omega 1 round integrality gaps were known integrality gaps for the 2 rightarrow 4 norm had previously been sought due to its connection to small set expansion sse and unique games ug in both cases no go theorems were previously known based on computational assumptions such as the exponential time hypothesis eth which asserts that 3 sat requires exponential time to solve our unconditional results achieve the same parameters as all of these previous results for separable states or as some of the previous results for quantum correlations in some cases we can make use of the framework of lee raghavendra steurer lrs to establish integrality gaps for any sdp extended formulation not only the sos hierarchy our hardness result on separable states also yields a dimension lower bound of approximate disentanglers answering a question of watrous and aaronson et al these results can be viewed as limitations on the monogamy principle the ppt test the ability of tsirelson type bounds to restrict quantum correlations as well as the sdp hierarchies of doherty parrilo spedalieri navascues pironio acin and berta fawzi scholz indeed a wide range of past work in quantum information can be described as using an sdp on one of the above two problems and our results put broad limits on these lines of argument

minimizer extraction in polynomial optimization is robust in this article we present a robustness analysis of the extraction of optimizers in polynomial optimization optimizers can be extracted by solving moment problems using flatness and the gelfand n

free pick functions representations asymptotic behavior and matrix monotonicity in several noncommuting variables r nevanlinna showed that elements of the pick class the set of complex analytic functions taking the upper half plane into itself have certain integral representations which reflect their asymptotic behavior at infinity later lowner connected the pick class to matrix monotone functions we generalize the nevanlinna representation theorems and lowner s theorem on matrix monotone functions to the free pick class the collection of functions that map tuples of matrices with positive imaginary part into the matrices with positive imaginary part which obey the free functional calculus

monotonic properties of the least squares mean we settle an open problem of several years standing by showing that the least squares mean for positive definite matrices is monotone for the usual loewner order indeed we show this is a special case of its appropriate generalization to partially ordered complete metric spaces of nonpositive curvature our techniques extend to establish other basic properties of the least squares mean such as continuity and joint concavity moreover we introduce a weighted least squares mean and derive our results in this more general setting

noncommutative polynomials nonnegative on a variety intersect a convex set abstract this paper gives a precise algebraic certificate for a noncommutative polynomial to be nonnegative on a free convex semialgebraic set intersect the variety of a free left ideal

operator monotone functions and lowner functions of several variables we prove generalizations of lowner s results on matrix monotone functions to several variables we give a characterization of when a function of d variables is locally monotone on d tuples of commuting self adjoint n by n matrices we prove a generalization to several variables of nevanlinna s theorem describing analytic functions that map the upper half plane to itself and satisfy a growth condition we use this to characterize all rational functions of two variables that are operator monotone

noncommutative ball maps in this paper we analyze problems involving matrix variables for which we use a noncommutative algebra setting to be more specific we use a class of functions called nc analytic functions defined by power series in noncommuting variables and evaluate these functions on sets of matrices of all dimensions we call such situations dimension free these types of functions have recently been used in the study of dimension free linear system engineering problems in this paper we characterize nc analytic maps that send dimension free matrix balls to dimension free matrix balls and carry the boundary to the boundary such maps we call nc ball maps we find that up to normalization an nc ball map is the direct sum of the identity map with an nc analytic map of the ball into the ball that is nc ball maps are very simple in contrast to the classical result of d angelo on such analytic maps in c another mathematically natural class of maps carries a variant of the noncommutative distinguished boundary to the boundary but on these our results are limited we shall be interested in several types of noncommutative balls conventional ones but also balls defined by constraints called linear matrix inequalities lmi what we do here is a small piece of the bigger puzzle of understanding how lmis behave with respect to noncommutative change of variables

monotonicity of the matrix geometric mean an attractive candidate for the geometric mean of m positive definite matrices a1 am is their riemannian barycentre g one of its important operator theoretic properties monotonicity in the m arguments has been established recently by lawson and lim we give an elementary proof of this property using standard matrix analysis and some counting arguments we derive some new inequalities for g one of these says that for any unitarily invariant norm g is not bigger than the geometric mean of a1 am

fixed point theorems for noncommutative functions abstract we establish a fixed point theorem for mappings of square matrices of all sizes which respect the matrix sizes and direct sums of matrices the conclusions are stronger if such a mapping also respects matrix similarities i e is a noncommutative function as a special case we prove the corresponding contractive mapping theorem which can be viewed as a new version of the banach fixed point theorem this result is then applied to prove the existence and uniqueness of a solution of the initial value problem for odes in noncommutative spaces as a by product of the ideas developed in this paper we establish a noncommutative version of the principle of nested closed sets

factorization of operator valued polynomials in several non commuting variables abstract a version of fejer riesz factorization and factorization of positive operator valued polynomials in several non commuting variables holds the proofs use arveson s extension theorem and matrix completions

solving moment problems by dimensional extension polynomial inequalities the second part of the paper uses the representation results of positive functionals on certain spaces of rational functions developed in the flrst part for decomposing a polynomial which is positive on such a semi algebraic set into a canonical sum of squares of rational functions times explicit multipliers let n 1 be a flxed integer due to the fact that for n 1 not every nonnegative polynomial in r n can be written as a sum of squares of polynomials see for instance 2 x6 3 the moment problems in n variables are more di cult than the classical one variable problems this very intriguing territory has been investigated by many authors see 2 7 12 and their references although characterizations for measures whose support lies in an arbitrary generally unbounded semi algebraic set do not seem to exist the present paper starts from an idea of the second author see 19 about solving moment problems by a change of basis via an embedding of r n into a submanifold of a higher dimensional euclidean space rougly speaking we prove that certain n 1 dimensional extensions of a moment sequence are naturally characterized by positivity conditions and moreover these extensions parametrize all possible solutions of the moment problem to be more speciflc let fi z r n x fi d x fi 2 z n

connes embedding conjecture and sums of hermitian squares we show that connes embedding conjecture on von neumann algebras is equivalent to the existence of certain algebraic certificates for a polynomial in noncommuting variables to satisfy the following nonnegativity condition the trace is nonnegative whenever self adjoint contraction matrices of the same size are substituted for the variables these algebraic certificates involve sums of hermitian squares and commutators we prove that they always exist for a similar nonnegativity condition where elements of separable ii1 factors are considered instead of matrices under the presence of connes conjecture we derive degree bounds for the certificates

loewner s theorem in several variables abstract in this paper we establish a multivariable non commutative generalization of loewner s theorem characterizing operator monotone functions as real functions admitting analytic continuation mapping the upper complex half plane into itself the non commutative several variable theorem proved here represents several variable operator monotone functions of positive operator tuples not a priori assumed to be analytic or continuous as a schur complement of a positive linear matrix pencil to achieve this we establish the abstract integral formula for these functions using only matrix convexity and lmis that represents operator monotone and operator concave free functions as a conditional expectation of a schur complement of a linear matrix pencil on a tensor product operator algebra the results can be applied to any of the various multivariable operator means that have been constructed in the last decades or so including the karcher mean to obtain an explicit closed formula for these operator means

statistical convergence and ideal convergence for sequences of functions abstract let i p n stand for an ideal containing finite sets we discuss various kinds of statistical convergence and i convergence for sequences of functions with values in r or in a metric space for real valued measurable functions defined on a measure space x m we obtain a statistical version of the egorov theorem when x we show that in its assertion equi statistical convergence on a big set cannot be replaced by uniform statistical convergence also we consider statistical convergence in measure and i convergence in measure with some consequences of the riesz theorem we prove that outer and inner statistical convergences in measure for sequences of measurable functions are equivalent if the measure is finite

a measure theoretical subsequence characterization of statistical convergence the concept of statistical convergence of a sequence was first introduced by h fast statistical convergence was generalized by r c buck and studied by other authors using a regular nonnegative summability matrix a in place of c1 the main result in this paper is a theorem that gives meaning to the statement s s converges to l statistically t if and only if most of the subsequences of s converge in the ordinary sense to l here t is a regular nonnegative and triangular matrix corresponding results for lacunary statistical convergence recently defined and studied by j a fridy and c orhan are also presented introduction the concept of the statistical convergence of a sequence of reals s sn was first introduced by h fast 9 the sequence s sn is said to converge statistically to l and we write lim s l stat if for every e n oo rim n k e1 0 where jai denotes the cardinality of the set a properties of statistically convergent sequences were studied in 5 6 12 and 16 in 13 fridy and miller gave a characterization of statistical convergence for bounded sequences using a family of matrix summability methods statistical convergence can be generalized by using a regular nonnegative summability matrix a in place of c this idea was first mentioned by r c buck 3 in 1953 and has been further studied by sember and freedman 10 and 11 and connor 5 and 7 regular nonnegative summability matrices turn out to be too general for our purposes here instead we use the concept of a mean a matrix t amn will be called a mean if amn 0 when n m e iamn 1 for all m and limm 0oam 0 for each n if t amn is a mean following buck a sequence s sn is said to be statistically t summable to l and we write sn l stat t if for every e 0 received by the editors august 18 1993 and in revised form february 14 1994 originally communicated to the proceedings of the ams by andrew bruckner 1991 mathematics subject classification primary 40d25 secondary 40g99 28a12 1995 american mathematical society 0002 9947 95 1 00 s 25 per page

positive noncommutative polynomials are sums of squares hilbert s 17th problem concerns expression of polynomials on r n as a sum of squares it is well known that many positive polynomials are not sums of squares see re d a for excellent surveys in this paper we consider symmetric noncommutative polynomials and call one matrix positive if whenever matrices of any size are substituted for the variables in the polynomial the matrix value which the polynomial takes is positive semidefinite the result in this paper is a polynomial is matrix positive if and only if it is a sum of squares

every convex free basic semi algebraic set has an lmi representation the matricial solution set of a linear matrix inequality lmi is a convex free basic open semi algebraic set the main theorem of this paper is a converse each such set arises from some lmi the result has implications for semi definite programming and systems engineering as well as for free semi algebraic geometry

extensions of positive definite functions on free groups abstract an analogue of krein s extension theorem is proved for operator valued positive definite functions on free groups the proof gives also the parametrization of all extensions by means of a generalized type of szego parameters one singles out a distinguished completion called central which is related to quasi multiplicative positive definite functions an application is given to factorization of noncommutative polynomials

constrained polynomial optimization problems with noncommuting variables in this paper we study constrained eigenvalue optimization of noncommutative nc polynomials focusing on the polydisc and the ball our three main results are as follows 1 an nc polynomial is nonnegative if and only if it admits a weighted sum of hermitian squares decomposition 2 eigenvalue optima for nc polynomials can be computed using a single semidefinite program sdp this sharply contrasts with the commutative case where sequences of sdps are needed 3 the dual solution to this single sdp can be exploited to extract eigenvalue optimizers with an algorithm based on two ingredients solution to a truncated nc moment problem via flat extensions and gelfand naimark segal construction the implementation of these procedures in our computer algebra system ncsostools is presented and several examples pertaining to matrix inequalities are given to illustrate our results

semigroup c algebras and amenability of semigroups we construct reduced and full semigroup c algebras for left cancellative semigroups our new construction covers particular cases already considered by a nica and also toeplitz algebras attached to rings of integers in number fields due to j cuntz moreover we show how left amenability of semigroups can be expressed in terms of these semigroup c algebras in analogy to the group case

operator positivstellensatze for noncommutative polynomials positive on matrix convex sets abstract this article studies algebraic certificates of positivity for noncommutative nc operator valued polynomials on matrix convex sets such as the solution set d l called a free hilbert spectrahedron of the linear operator inequality loi l x a 0 i j 1 g a j x j 0 where a j are self adjoint linear operators on a separable hilbert space x j matrices and i is an identity matrix if a j are matrices then l x 0 is called a linear matrix inequality lmi and d l a free spectrahedron for monic lmis i e a 0 i and nc matrix valued polynomials the certificates of positivity were established by helton klep and mccullough in a series of articles with the use of the theory of complete positivity from operator algebras and classical separation arguments from real algebraic geometry since the full strength of the theory of complete positivity is not restricted to finite dimensions but works well also in the infinite dimensional setting we use it to tackle our problems first we extend the characterization of the inclusion d l 1 d l 2 from monic lmis to monic lois l 1 and l 2 as a corollary one immediately obtains the description of a polar dual of a free hilbert spectrahedron d l and its projection called a free hilbert spectrahedrop further on using this characterization in a separation argument we obtain a certificate for multivariate matrix valued nc polynomials f positive semidefinite on a free hilbert spectrahedron defined by a monic loi replacing the separation argument by an operator fejer riesz theorem enables us to extend this certificate in the univariate case to operator valued polynomials f finally focusing on the algebraic description of the equality d l 1 d l 2 we remove the assumption of boundedness from the description in the lmis case by an extended analysis however the description does not extend to lois case by counterexamples