equivalence of hardy submodules generated by polynomials abstract in this paper we obtain a complete classification under unitary equivalence for hardy submodules on the polydisk which are generated by ideals of polynomials let i be an ideal of polynomials in n variables since i is generated by finitely many polynomials i has a greatest common divisor p so i can be uniquely written as i p l which is called the beurling form of i let i 1 p 1 l 1 i 2 p 2 l 2 we prove that i 1 and i 2 are unitarily equivalent if and only if there are polynomials q 1 and q 2 with z q 1 d n z q 2 d n such that p 1 q 1 p 2 q 2 on t n and p 1 l 1 p 1 l 2 consequently two principal submodules p 1 and p 2 are unitarily equivalent if and only if there are polynomials q 1 and q 2 with z q 1 d n z q 2 d n such that p 1 q 1 p 2 q 2 on t n furthermore we give a complete similarity classification for submodules generated by homogeneous ideals finally we point out that in the case of the hardy module on the unit ball i 1 and i 2 are unitarily equivalent if and only if they are equal if i 1 and i 2 are homogeneous ideals then i 1 and i 2 are quasi similar if and only if i 1 i 2

essentially normal homogeneous quotient modules on the polydisc abstract in the present paper we give a complete characterization of all essentially normal homogeneous quotient modules of the hardy space on the unit polydisc it turns out that our approach also applies to quotients of weighted bergman modules

invariant subspaces dilation theory and the structure of the predual of a dual algebra i on etudie la structure des preduaux d algebres d operateurs on obtient un nouveau theoreme de sous espace invariant on trouve des proprietes concernant le treillis sous espace invariant de l operateur decalage de bergman

on quotient modules the case of arbitrary multiplicity let m be a hilbert module of holomorphic functions defined on a bounded domain cm let m0 be the submodule of functions vanishing to order k on a hypersurface z in in this paper we describe the quotient module mq

characteristic spaces and rigidity for analytic hilbert modules abstract it is well known that a polynomial in one variable is completely determined by its zeros counting multiplicities we generalize this result to an ideal of polynomials in several variables by introducing the characteristic spaces of the ideal one finds that the ideal is completely determined by its characteristic spaces on a characteristic set in particular a primary ideal is completely determined by its characteristic space at any zero point some straightforward applications of the above results yield the algebraic reduction theorem for analytic hilbert modules in several variables also we obtain some general rigidity results for analytic hilbert modules by using the techniques of af envelopes of analytic hilbert modules

subalgebras of c algebras iii multivariable operator theory a d contraction is a d tuple t1 td of mutually commuting operators acting on a common hilbert space h such that t1 1 t2 2 td d 1 2 d for all 1 2 d h these are the higher dimensional counterparts of contractions we show that many of the operator theoretic aspects of function theory in the unit disk generalize to the unit ball bd in complex d space including von neumann s inequality and the model theory of contractions these results depend on properties of the d shift a distinguished d contraction which acts on a new h2 space associated with bd and which is the higher dimensional counterpart of the unilateral shift h 2 and the d shift are highly unique indeed by exploiting the noncommutative choquet boundary of the d shift relative to its generated c algebra we find that there is more uniqueness in dimension d 2 than there is in dimension one

a characterization of operator algebras abstract an operator algebra is a uniformly closed algebra of bounded operators on a hilbert space in this paper we give a characterization of unital operator algebras in terms of their matricial norm structure more precisely if a is an l matricially normed space and also an algebra with a completely contractive multiplication and an identity of norm 1 then there is a completely isometric isomorphism of a onto a unital operator algebra indeed the multiplication on a need not be assumed to be associative for this conclusion to follow examples are given to show that the condition on the identity is necessary it follows from the above that the quotient of an operator algebra by a closed two sided ideal with the natural matricial structure is again an operator algebra up to complete isometric isomorphism

essential normality of polynomial generated submodules hardy space and beyond abstract recently douglas and wang proved that for each polynomial q the submodule q of the bergman module generated by q is essentially normal 11 using improved techniques we show that the hardy space analogue of this result holds and more

complete quotient boolean algebras for i a proper countably complete ideal on the power set 9 x for some set x can the quotient boolean algebra 9 x i be complete we first show that if the cardinality of x is at least w3 then having completeness implies the existence of an inner model with a measurable cardinal a well known situation that entails completeness is when the ideal i is a nontrivial ideal over a cardinal k which is k saturated the second author had established the sharp result that it is consistent by forcing to have such an ideal over k co relative to the existence of a woodin cardinal augmenting his proof by interlacing forcings that adjoin boolean suprema we establish relative to the same large cardinal hypothesis the consistency of 2w1 wt3 and there is an ideal ideal i over co such that y cwl i is complete the cardinality assertion implies that there is no ideal over co which is w02 saturated and so completeness of the boolean algebra and saturation of the ideal has been separated for i a proper countably complete ideal on 92 x for some set x can the quotient boolean algebra 9 x i be complete this question was raised by sikorski si in 1949 by a simple projection argument as for measurable cardinals it can be assumed that x is an uncountable cardinal k and that i is a k complete ideal on 9z k containing all singletons in this paper we provide consequences from and consistency results about completeness throughout k will denote an uncountable cardinal and by an ideal over k we shall mean a proper k complete ideal on 9z k containing all singletons if k is a measurable cardinal and i a prime ideal over k then of course k9 k i is complete being the two element boolean algebra the following theorem shows that completeness in itself has strong consistency strength theorem a if k 03 and there is an ideal i over k such that 9 k i is complete then there is an inner model with a measurable cardinal the restriction k 03 is necessary for our proof there is a well known situation in which completeness obtains an ideal over k is a saturated iff whenever xaia ai c 9 k i there are a 8 a such that x n xy i smith tarski st established that if an ideal i over k is k saturated then 9f k i is complete there are several consistency results about the existence of such ideals modifying a sharp one due to the second author we establish the following result separating completeness from saturation received by the editors august 13 1990 and in revised form january 28 1994 1991 mathematics subject classification primary 03e35 secondary 03e55 the second author is partially supported by the u s israel binational science foundation this paper is 390 in his bibliography 1995 american mathematical society 0002 9947 95 1 00 25 per page

operator theory in the hardy space over the bidisk iii this paper is a continuation of a project of developing a systematic operator theory inh2 d2 a large part of it is devoted to a study ofevaluation operator which is a very useful tool in the theory a number of elementary properties of the evaluation operator are exhibited and these properties are used to derive results in other topics such as interpretation of characteristic opertor function inh2 d2 spectral equivalence compactness compressions of shift operators etc even though some results reflect the two variable nature ofh2 d2 the goal of this paper is to manifest a close tie between the operator theory inh2 d2 and classical single operator theory the unilateral shift of a finite multiplicity and the bergman shift will be used as examples to illustrate some of the results

ordered products w algebra and two variable definite parity orthogonal polynomials it has been shown that the cartan subalgebra of w algebra is the space of the two variable definite parity polynomials explicit expressions of these polynomials and their basic properties are presented it also has been shown that they carry the infinite dimensional irreducible representation of the su 1 1 algebra having the spectrum bounded from below a realization of this algebra in terms of difference operators is also obtained for particular values of the ordering parameter s they are identified with the classical orthogonal polynomials of a discrete variable such as the meixner meixner pollaczek and askey wilson polynomials with respect to variable s they satisfy a second order eigenvalue equation of hypergeometric type exact scattering states with zero energy for a family of potentials are expressed in terms of these polynomials it has been put forward that it is the inonu wigner contraction and its inverse that form a bridge between the difference and differential calculus

spectral truncations in noncommutative geometry and operator systems in this paper we extend the traditional framework of noncommutative geometry in order to deal with spectral truncations of geometric spaces i e imposing an ultraviolet cutoff in momentum space and with tolerance relations which provide a coarse grain approximation of geometric spaces at a finite resolution in our new approach the traditional role played by c algebras is taken over by operator systems as part of the techniques we treat c envelopes dual operator systems and stable equivalence we define a propagation number for operator systems which we show to be an invariant under stable equivalence and use to compare approximations of the same space we illustrate our methods for concrete examples obtained by spectral truncations of the circle these are operator systems of finite dimensional toeplitz matrices and their dual operator systems which are given by functions in the group algebra on the integers with support in a fixed interval it turns out that the cones of positive elements and the pure state spaces for these operator systems possess a very rich structure which we analyze including for the algebraic geometry of the boundary of the positive cone and the metric aspect i e the distance on the state space associated to the dirac operator the main property of the spectral truncation is that it keeps the isometry group intact in contrast if one considers the other finite approximation provided by circulant matrices the isometry group becomes discrete even though in this case the operator system is a c algebra we analyze this in the context of the finite fourier transform on the cyclic group the extension of noncommutative geometry to operator systems allows one to deal with metric spaces up to finite resolution by considering the relation d x y varepsilon between two points or more generally a tolerance relation which naturally gives rise to an operator system

a harmonic analysis approach to essential normality of principal submodules guo and the second author have shown that the closure i in the drury arveson space of a homogeneous principal ideal i in c z1 zn is essentially normal in this note the authors extend this result to the closure of any principal polynomial ideal in the bergman space in particular the commutators and cross commutators of the restrictions of the multiplication operators are shown to be in the schatten p class for p n the same is true for modules generated by polynomials with vector valued coefficients further the maximal ideal space xi of the resulting c algebra for the quotient module is shown to be contained in z i bn where z i is the zero variety for i and to contain all points in bn that are limit points of z i bn finally the techniques introduced enable one to study a certain class of weight bergman spaces on the ball

beurling type quotient modules over the bidisk and boundary representations this paper mainly concerns beurling type quotient modules of h2 d2 over the bidisk by establishing a theorem of function theory over the bidisk it is shown that a beurling type quotient module is essentially normal if and only if the corresponding inner function is a rational inner function having degree at most 1 1 furthermore we apply this result to the study of boundary representations of toeplitz algebras over quotient modules it is proved that the identity representation of c sz sw is a boundary representation of b sz sw in all nontrivial cases this extends a result of arveson to toeplitz algebras on beurling type quotient modules over the bidisk cf w arveson subalgebras of c algebras acta math 123 1969 141 224 w arveson subalgebras of c algebras ii acta math 128 1972 271 308 the paper also establishes k homology defined by beurling type quotient modules over the bidisk

martin s maximum saturated ideals and non regular ultrafilters part ii we prove assuming the existence of a huge cardinal the consistency of fully non regular ultrafilters on the successor of any regular cardinal we also construct ultrafilters with ultraproducts of small cardinality part ii is logically independent of part i

hilbert polynomials and arveson s curvature invariant abstract we define and study hilbert polynomials for certain holomorphic hilbert spaces we obtain several estimates for these polynomials and their coefficients our estimates inspire us to investigate the connection between the leading coefficients of hilbert polynomials for invariant subspaces of the symmetric fock space and arveson s curvature invariant for coinvariant subspaces we are able to obtain some formulas relating the curvature invariant with other invariants in particular we prove that arveson s version of the gauss bonnet chern formula is true when the invariant subspaces are generated by any polynomials

the isomorphism problem for some universal operator algebras abstract this paper addresses the isomorphism problem for the universal non self adjoint operator algebras generated by a row contraction subject to homogeneous polynomial relations we find that two such algebras are isometrically isomorphic if and only if the defining polynomial relations are the same up to a unitary change of variables and that this happens if and only if the associated subproduct systems are isomorphic the proof makes use of the complex analytic structure of the character space together with some recent results on subproduct systems restricting attention to commutative operator algebras defined by a radical ideal of relations yields strong resemblances with classical algebraic geometry these commutative operator algebras turn out to be algebras of analytic functions on algebraic varieties we prove a projective nullstellensatz connecting closed ideals and their zero sets under some technical assumptions we find that two such algebras are isomorphic as algebras if and only if they are similar and we obtain a clear geometrical picture of when this happens this result is obtained with tools from algebraic geometry reproducing kernel hilbert spaces and some new complex geometric rigidity results of independent interest the c envelopes of these algebras are also determined the banach algebraic and the algebraic classification results are shown to hold for the wot closures of these algebras as well

analytic models for commuting operator tuples on bounded symmetric domains for a domain in c d and a hilbert space h of analytic functions on which satisfies certain conditions we characterize the commuting d tuples t t 1 t d of operators on a separable hilbert space h such that t is unitarily equivalent to the restriction of m to an invariant subspace where m is the operator d tuple z i on the hilbert space tensor product h h for the unit disc and h the hardy space h 2 this reduces to a well known theorem of sz nagy and foias for h a reproducing kernel hilbert space on c c d such that the reciprocal 1 k x y of its reproducing kernel is a polynomial in x and y this is a recent result of ambrozie muller and the second author in this paper we extend the latter result by treating spaces h for which 1 k ceases to be a polynomial or even has a pole namely the standard weighted bergman spaces or rather their analytic continuation h h on a cartan domain corresponding to the parameter v in the continuous wallach set and reproducing kernel hilbert spaces h for which 1 k is a rational function further we treat also the more general problem when the operator m is replaced by m w w being a certain generalization of a unitary operator tuple for the case of the spaces h on cartan domains our results are based on an analysis of the homogeneous multiplication operators on which seems to be of an independent interest

compressions of compact tuples abstract we study the matrix range of a tuple of compact operators on a hilbert space and examine the notions of minimal nonsingular and fully compressed tuples in this pursuit we refine previous results by characterizing nonsingular compact tuples in terms of matrix extreme points of the matrix range further we find that a compact tuple a is fully compressed if and only if it is multiplicity free and the shilov ideal is trivial which occurs if and only if a is minimal and nonsingular fully compressed compact tuples are therefore uniquely determined up to unitary equivalence by their matrix ranges we also produce a proof of this fact which does not depend on the concept of nonsingularity

free field representation for massive integrable models a new approach to massive integrable models is considered it allows one to find symmetry algebras which define the spaces of local operators and to get general integral representations for form factors in thesu 2 thirring and sine gordon models

the kirchberg wassermann operator system is unique in 1998 kirchberg and wassermann constructed a separable nuclear operator system with the property that the canonical unital homomorphism from the maximal c algebra onto the c envelope is injective we show that such an operator system is unique and completely order isomorphic to the operator system associated with the noncommutative poulsen simplex

confluent hypergeometric orthogonal polynomials related to the rational quantum calogero system with harmonic confinement two families type a and type b of confluent hypergeometric polynomials in several variables are studied we describe the orthogonality properties differential equations and pieri type recurrence formulas for these families in the one variable case the polynomials in question reduce to the hermite polynomials type a and the laguerre polynomials type b respectively the multivariable confluent hypergeometric families considered here may be used to diagonalize the rational quantum calogero models with harmonic confinement for the classical root systems and are closely connected to the symmetric generalized spherical harmonics investigated by dunkl

the quantum q krawtchouk and q meixner polynomials and their related d functions for the quantum groups su q 2 and su q 1 1 the complete comparative analysis of the quantum q krawtchouk and q meixner polynomials of a discrete variable on a nonuniform grid x s q2s and the d function for the quantum groups suq 2 and suq 1 1 is done the complete set of characteristics of these polynomials i e orthogonality relations normalization factors recurrent relations the explicit analytical expressions the rodrigues formulas the formulas of difference derivatives various particular values and cases are calculated the correlations between the properties of the polynomials mentioned above and the d functions for the quantum groups suq 2 and suq 1 1 are established in the case of suq 1 1 only d functions for the positive discrete series of the unitary irreducible representations are considered it is known that on the nonuniform grid x s q2s there are two kinds of krawtchouk and meixner polynomials also the properties of the second kind of these polynomials which are not connected to the d functions are discussed

restrictions of h p functions in the polydisk manuscript received 19 may 1987 this work was partially supported by an n s f grant 2 added in proof as r rochberg and n salinas have pointed out to me this actually goes back to n aronszajn trans amer math soc 68 1950 p 361 where it was proved that the product kl xl yl k x y2 of reproducing kernels for spaces fl f corresponds to the tensor product space f f while the product kl x y k x y corresponds to the restriction of f f to the diagonal americant journial of mathematics 110 1988 1119 1152 1119

essential normality essential norms and hyperrigidity abstract let s s 1 s d denote the compression of the d shift to the complement of a homogeneous ideal i of c z 1 z d arveson conjectured that s is essentially normal in this paper we establish new results supporting this conjecture and connect the notion of essential normality to the theory of the c envelope and the noncommutative choquet boundary the unital norm closed algebra b i generated by s 1 s d modulo the compact operators is shown to be completely isometrically isomorphic to the uniform algebra generated by polynomials on v z i b d where z i is the variety corresponding to i consequently the essential norm of an element in b i is equal to the sup norm of its gelfand transform and the c envelope of b i is identified as the algebra of continuous functions on v b d which means it is a complete invariant of the topology of the variety determined by i in the ball motivated by this determination of the c envelope of b i we suggest a new more qualitative approach to the problem of essential normality we prove the tuple s is essentially normal if and only if it is hyperrigid as the generating set of a c algebra which is a property closely connected to arveson s notion of a boundary representation we show that most of our results hold in a much more general setting in particular for most of our results the ideal i can be replaced by an arbitrary not necessarily homogeneous invariant subspace of the d shift

the reducibility of compressed shifts on beurling type quotient modules over the bidisk abstract in this paper we study the compressed shift operator s z 1 on the beurling type quotient module k of hardy space h 2 d 2 over the bidisk firstly we give a necessary and sufficient condition such that s z 1 has nontrivial pure isometry reducing subspace as an application we show that s z 1 has agler reducing subspaces if and only if is the product of two one variable inner functions secondly for a rational inner function with degree n 1 we show that s z 1 is reducible on k if and only if s z 1 has agler reducing subspaces furthermore we study the case when the rational inner functions have degree n 2 and this case is quite different from that the degree of is n 1

matrix convex sets without absolute extreme points abstract this article shows the existence of a class of closed bounded matrix convex sets which do not have absolute extreme points the sets we consider are noncommutative sets k x formed by taking matrix convex combinations of a single tuple x in the case that x is a tuple of compact operators with no nontrivial finite dimensional reducing subspaces k x is a closed bounded matrix convex set with no absolute extreme points a central goal in the theory of matrix convexity is to find a natural notion of an extreme point in the dimension free setting which is minimal with respect to spanning matrix extreme points are the strongest type of extreme point known to span matrix convex sets however they are not necessarily the smallest set which does so absolute extreme points a more restricted type of extreme points that are closely related to arveson s boundary enjoy a strong notion of minimality should they span this result shows that matrix convex sets may fail to be spanned by their absolute extreme points

the dirac operator of a commuting d tuple abstract given a commuting d tuple t t1 td of otherwise arbitrary operators on a hilbert space there is an associated dirac operator dt significant attributes of the d tuple are best expressed in terms of dt including the taylor spectrum and the notion of fredholmness in fact all properties of t derive from its dirac operator we introduce a general notion of dirac operator in dimension d 1 2 that is appropriate for multivariable operator theory we show that every abstract dirac operator is associated with a commuting d tuple and that two dirac operators are isomorphic iff their associated operator d tuples are unitarily equivalent by relating the curvature invariant introduced in a previous paper to the index of a dirac operator we establish a stability result for the curvature invariant for pure d contractions of finite rank it is shown that for the subcategory of all such t that are a fredholm and and b graded the curvature invariant k t is stable under compact perturbations we do not know if this stability persists when t is fredholm but ungraded although there is concrete evidence that it does

an analytic grothendieck riemann roch theorem we extend the boutet de monvel toeplitz index theorem to complex manifolds with isolated singularities following the relative k homology theory of baum douglas and taylor for manifolds with boundary we apply this index theorem to study the arveson douglas conjecture let bmbm be the unit ball in cmcm and i an ideal in the polynomial algebra c z1 zm c z1 zm we prove that when the zero variety zizi is a complete intersection space with only isolated singularities and intersects with the unit sphere s2m 1s2m 1 transversely the representations of c z1 zm c z1 zm on the closure of i in la2 bm and also the corresponding quotient space qiqi are essentially normal furthermore we prove an index theorem for toeplitz operators on qiqi by showing that the representation of c z1 zm c z1 zm on the quotient space qiqi gives the fundamental class of the boundary zi s2m 1zi s2m 1 in the appendix we prove with kai wang that if f la2 bm vanishes on zi bmzi bm then f is contained inside the closure of the ideal i in la2 bm

multiplier algebras of complete nevanlinna pick spaces dilations boundary representations and hyperrigidity abstract we study reproducing kernel hilbert spaces on the unit ball with the complete nevanlinna pick property through the representation theory of their algebras of multipliers we give a complete description of the representations in terms of the reproducing kernels while representations always dilate to representations of the ambient c algebra we show that in our setting we automatically obtain coextensions in fact we show that in many cases this phenomenon characterizes the complete nevanlinna pick property we also deduce operator theoretic dilation results which are in the spirit of work of agler and several other authors moreover we identify all boundary representations compute the c envelopes and determine hyperrigidity for certain analogues of the disc algebra finally we extend these results to spaces of functions on homogeneous subvarieties of the ball

blaschke products and the rank of backward shift invariant subspaces over the bidisk abstract let h 2 d 2 be the hardy space over the bidisk for sequences of blaschke products n z n and n w n satisfying some additional conditions we may define a rudin type invariant subspace m we shall determine the rank of h 2 d 2 m for the pair of operators t z and t w

the hahn and meixner polynomials of an imaginary argument and some of their applications the hahn and meixner polynomials belonging to the classical orthogonal polynomials of a discrete variable are analytically continued in the complex plane both in variable and parameter this leads to the origination of two systems of real polynomials orthogonal with respect to a continuous measure the meixner polynomials of an imaginary argument obtained in this manner turned out to be known in the literature as the pollaczek polynomials the orthogonality relation for the hahn polynomials with respect to a continuous measure is apparently new a close connection between the hahn polynomials of an imaginary argument and representations of the lorentz group so 3 1 is considered

toeplitz c algebras on bounded symmetric domains in this paper we develop a structure theory for the toeplitz c algebra y for an arbitrary bounded symmetric domain d in c classical or exceptional as the main result it is shown that 5y is a c algebra of type i which is solvable i e admits a finite composition sequence of closed ideals sk in 7 such that the subquotients fk i k are continuous fields of elementary c algebras of compact operators the length of this composition sequence coincides with the rank r of d as a hermitian symmetric space for 0 k r the spectral components sk of ay parametrizing the irreducible representations of y are realized in terms of the boundary components complex analytic faces of d the structure theory for y is based on the fact that the tangent space z c c of d carries a jordan algebraic structure z is a jordan triple system with triple product uv w describing the geometry of d in algebraic terms for example d is the open unit ball with respect to the so called spectral norm on z and the boundary components of d are in 1 1 correspondence with the tripotents e e z partial isometries satisfying ee e 4 e in particular the shilov boundary s of d consists of all tripotents of maximal rank r the jordan theoretic description of d was used in 17 to determine an explicit peter weyl decomposition of the hardy space h2 s cf 15 in terms of the norm functions generalized determinants associated with the jordan triple

properties of beurling type submodules via agler decompositions abstract in this paper we study operator theoretic properties of the compressed shift operators s z 1 and s z 2 on complements of submodules of the hardy space over the bidisk h 2 d 2 specifically we study beurling type submodules namely submodules of the form h 2 d 2 for inner using properties of agler decompositions of to deduce properties of s z 1 and s z 2 on model spaces h 2 d 2 h 2 d 2 results include characterizations in terms of of when a commutator s z j s z j has rank n and when subspaces associated to agler decompositions are reducing for s z 1 and s z 2 we include several open questions

pure matrix states on operator systems abstract an operator system is a complex matricially ordered vector space that is completely order isomorphic to a unital selfadjoint subspace of a unital c algebra a matrix state on an operator system v is a unital completely positive linear map of v into a full matrix algebra pure matrix states are studied and a new and somewhat simplified proof of a krein milman type theorem of webster and winkler is given if v is 3 dimensional then the matrix state space of v is matrix affinely homeomorphic to the matricial range of some hilbert space operator with the aid of this representation pure matrix states on 3 dimensional operator systems are examined and in some cases completely determined

essential normality for quotient modules and complex dimensions abstract for analytic sets m introduced in 9 we show that the corresponding quotient module q of the bergman module is p essentially normal for all p dim c m which verifies the geometric arveson douglas conjecture in this case this result makes it possible to study the helton howe trace invariants on both q and the corresponding submodule r

multivariable continuous hahn and wilson polynomials related to integrable difference systems multivariable generalizations of the continuous hahn and wilson polynomials are introduced as eigenfunctions of rational ruijsenaars type difference systems with an external field

operator algebras of monomial ideals in noncommuting variables abstract we study operator algebras arising from monomial ideals in the ring of polynomials in noncommuting variables through the apparatus of subproduct systems and c correspondences we provide a full comparison amongst the related operator algebras for our analysis we isolate a partially defined dynamical system to which we refer as the quantised dynamics of the monomial ideal in addition we revisit several previously considered constructions these include matsumoto s subshift c algebras as well as the tensor and the pimsner algebras associated with dynamical systems or graphs we sort out the various relations by giving concrete conditions and counterexamples that orientate the operator algebras of our context it appears that the boundary c algebras do not arise as the quotient with the compact operators unconditionally we establish a dichotomy to this effect by examining the resulting tensor algebras we identify their boundary representations we analyse their c envelopes and we give criteria for hyperrigidity moreover we completely classify them in terms of the data provided by the monomial ideals for tensor algebras of c correspondences and bounded isomorphisms this is achieved up to the level of local conjugacy in the sense of davidson and roydor for the quantised dynamics for tensor algebras of subproduct systems and algebraic isomorphisms this is achieved up to the level of equality of monomial ideals modulo permutations of the variables in the process we accomplish more in different directions most notably we show that tensor algebras form a complete invariant for isomorphic resp similar subproduct systems of homogeneous ideals up to isometric resp bounded isomorphisms the results on local conjugacy are obtained via an alternative proof of the breakthrough result of davidson and katsoulis on piecewise conjugate systems for our purposes we use appropriate compressions of the fock representation we then apply this alternative proof locally for the partially defined quantised dynamics in this way we avoid the topological graphs machinery and pave the way for further applications these include operator algebras of dynamical systems over commuting contractions or over row commuting contractions

quotient modules for some hilbert modules over the bidisk in this note we will prove the essential normality of some type of quotient hilbert modules over the bidisk moreover we will study some properties of the compression of the multiplication operators to the quotient hilbert modules

q analogue of the krawtchouk and meixner orthogonal polynomials abstract the comparative analysis of krawtchouk polynomials on a uniform grid with wigner d functions for the su 2 group is done as a result the partnership between corresponding properties of the polynomials and d functions is established giving the group theoretical interpretation of the properties of the krawtchouk polynomials in order to extend such an analysis to the quantum groups suq 2 and suq 1 1 q analogues of krawtchouk and meixner polynomials of a discrete variable are studied in detail as solutions of the finite difference equation of hypergeometric type on the nonuniform grid x s q2s however on this grid there are two kinds of krawtchouk and meixner polynomials characterized by different weight functions in the orthogonality relation as a first step to such analysis the simpler polynomials of the first kind are considered the total set of characteristics of these polynomials orthogonality condition normalization factor recurrent relation the explicit analytic expression the rodrigues formula the difference derivative formula various particular cases and values is calculated

samuel multiplicity and the structure of semi fredholm operators abstract two numerical invariants refining the fredholm index are introduced for any semi fredholm operator in such a way that their difference calculates the fredholm index these two invariants are inspired by samuel multiplicity in commutative algebra and can be regarded as the stabilized dimension of the kernel and cokernel a geometric interpretation of these invariants leads naturally to a 4 4 uptriangular matrix model for any semi fredholm operator on a separable hilbert space this model can be regarded as a refined local version of the apostol s 3 3 triangular representation for arbitrary operators some classical results such as gohberg s punctured neighborhood theorem can be read off directly from our matrix model banach space operators are also considered

geometric arveson douglas conjecture abstract we prove the arveson douglas essential normality conjecture for graded hilbert submodules that consist of functions vanishing on a given homogeneous subvariety of the ball smooth away from the origin our main tool is the theory of generalized toeplitz operators of boutet de monvel and guillemin

essentially normal hilbert modules and k homology this paper mainly concerns the essential normality of graded submodules essentially all of the basic hilbert modules that have received attention over the years are p essentially normal including the d shift hilbert module the hardy and bergman modules of the unit ball arveson conjectured graded submodules over the unit ball inherit this property and provided motivations to seek an affirmative answer some positive results have been obtained by arveson and douglas however the problem has been resistant in dimensions d 2 3 this paper shows that the arveson s conjecture is true in any dimension the paper also gives an affirmative answer in the case of the graded principal submodule finally the paper is associated with k homology invariants arising from graded quotient modules by which geometry of the quotient modules and geometry of algebraic varieties are connected in dimensions d 2 3 it is shown that k homology invariants determined by graded quotients are nontrivial the paper also establishes results on p smoothness of k homology elements and gives an explicit expression for k homology invariant in dimension d 2

the structure of inner multipliers on spaces with complete nevanlinna pick kernels let k be the reproducing kernel for a hilbert space h k of analytic functions on bd the open unit ball in cd d 1 k is called a complete np kernel if k0 1 and if 1 1 k z is positive definite on bd bd let d be a separable hilbert space and consider h k d h k d and think of it as a space of d valued h k functions a theorem of mccullough and trent j funct anal 178 2000 226 249 partially extends the beurling lax halmos theorem for the invariant subspaces of the hardy space h2 d they show that if k is a complete np kernel and if d is a separable hilbert space then for any scalar multiplier invariant subspace m of h k d there exists an auxiliary hilbert space e and a multiplication operator h k e h k d such that is a partial isometry and m h k e such multiplication operators are called inner multiplication operators and they satisfy the orthogonal projection onto m in this paper we shall show that for many interesting complete np kernels the analogy with the beurling lax halmos theorem can be strengthened we show that for almost every z bd the nontangential limit z of the multiplier bd b e d associated with an inner multiplication operator is a partial isometry and that rank z is equal to a constant almost everywhere the result applies to certain weighted dirichlet spaces and to the space h2d which is determined by the kernel k z 11 z d z bd in particular our result implies that the curvature invariant of arveson proc natl acad sci usa96 1999 11 096 11 099 of a pure contractive hilbert module of finite rank is an integer this answers a question of w arveson proc natl acad sci usa96 1999 11096 11099

quotients of standard hilbert modules we initiate a study of hilbert modules over the polynomial algebra that are obtained by completing with respect to an inner product having certain natural properties a standard hilbert module is a finite multiplicity version of one of these standard hilbert modules occupy a position analogous to that of free modules of finite rank in commutative algebra and their quotients by submodules give rise to universal solutions of nonlinear relations essentially all of the basic hilbert modules that have received attention over the years are standard including the hilbert module of the shift the hardy and bergman modules of the unit ball modules associated with more general domains in and those associated with projective algebraic varieties we address the general problem of determining when a quotient of an essentially normal standard hilbert module is essentially normal this problem has been resistant our main result is that it can be linearized in that the nonlinear relations defining the submodule can be reduced appropriately to linear relations through an iteration procedure and we give a concrete description of linearized quotients

fredholm and invertible tuples of operators the deformation problem using j l taylor s definition of joint spectrum we study fredholm and invertible tuples of operators on a hilbert space we give the foundations for a several variables theory including a natural generalization of atkinson s theorem and an index which well behaves we obtain a characterization of joint invertibility in terms of a single operator and study the main examples at length we then consider the deformation problem and solve it for the class of almost doubly commuting fredholm pairs with a semi fredholm coordinate 1 introduction 1 let t be a bounded linear operator on a banach space t is said to be invertible if there exists an operator s on such that ts st 1 the identity operator on 9c by the open mapping theorem this is equivalent to ker t 0 and r t range of t the last formulation does not rely upon the existence of an inverse for t but rather on the action of the operator t when t is replaced by an tuple of commuting operators several definitions of nonsingular ity exist j l taylor 19 has obtained one which reflects the actions of the operators by considering the koszul complex associated with the tuple 2 in this paper we develop a general several variables theory on the basis of taylor s work and study commuting and almost commuting commuting mod ulo the compacts tuples of operators on a hilbert space we obtain a characterization of joint invertibility in terms of the invertibility of a single operator which is essential for our approach from that we get a number of corollaries which generalize nicely the known elementary results in one variable at the same time the referred characterization allows us to define a continuous invariant under compact perturbations integer valued index on the class of fred holm tuples those almost commuting tuples which are invertible in the calkin algebra this index extends the classical one for fredholm operators we prove that an almost commuting tuple of essentially normal operators with all commu tators in trace class has index zero 2 and that a natural generalization of atkinson s theorem holds for tuples

dixmier trace for toeplitz operators on symmetric domains abstract for toeplitz operators on bounded symmetric domains of higher rank there is no obvious way to define the dixmier trace within the toeplitz c algebra since commutators are in general not compact in this paper we solve this problem by constructing a hilbert quotient module corresponding to partitions of length 1 which leads to commutators in the macaev class l n we also obtain an explicit formula for the dixmier trace of such commutators in terms of the underlying boundary geometry which involves a new type of flag manifold defined in jordan theoretic terms

the symmetric meixner pollaczek polynomials with real parameter abstract the symmetric meixner pollaczek polynomials p n x 2 2 for 0 are well studied polynomials these are polynomials orthogonal on the real line with respect to a continuous positive real measure for 0 p n x 2 2 are also polynomials however they are not orthogonal on the real line with respect to any real measure this paper defines a non standard inner product with respect to which the polynomials p n x 2 2 for 0 become orthogonal polynomials it examines the major properties of the polynomials p n x 2 2 for 0 which are also shared by the polynomials p n x 2 2 for 0

extremal problems inequalities and classical orthogonal polynomials in this survey paper we give a short account on characterizations for very classical orthogonal polynomials via extremal problems and the corresponding inequalities besides the basic properties of the classical orthogonal polynomials we consider polynomial inequalities of landau and kolmogoroff type some weighted polynomial inequalities in l 2 norm of markov bernstein type as well as the corresponding connections with the classical orthogonal polynomials

the noncommutative choquet boundary let s be an operator system a self adjoint linear sub space of a unital c algebra a such that 1 s and a c s is generated by s a boundary representation for s is an irreducible rep resentationof c s on a hilbert space with the property that s has a unique completely positive extension to c s the set s of all unitary equivalence classes of boundary representations is the non commutative counterpart of the choquet boundary of a function system s c x that separates points of x it is known that the closure of the choquet boundary of a function system s is the y boundary of x relative to s the corresponding noncommutative problem of whether every operator system has suf ficiently many boundary representations was formulated in 1969 but has remained unsolved despite progress on related issues in particu lar it was unknown if s 6 for generic s in this paper we show that every separable operator system has sufficiently many boundary representations our methods use separability in an essential way

toeplitz algebras subnormal tuples and rigidity on reproducing c z1 zd modules this paper mainly considers toeplitz algebras subnormal tuples and rigidity concerning reproducing c z1 zd modules by making use of arveson s boundary representation theory we find there is more rigidity in several variables than there is in single variable we specialize our attention to reproducing c z1 zd modules with u invariant kernels by examining the spectrum and the essential spectrum of the d tuple mz1 mzd and deducing an exact sequence of c algebras associated with toeplitz algebra finally we deal with toeplitz algebras defined on arveson submodules and rigidity of arveson submodules

star generating vectors of rudin s quotient modules abstract the purpose of this paper is to study a class of quotient modules of the hardy module h 2 d n along with the two variables quotient modules introduced by w rudin we introduce and study a large class of quotient modules namely rudin s quotient modules of h 2 d n by exploiting the structure of minimal representations we obtain an explicit co rank formula for rudin s quotient modules

residually finite dimensional operator algebras abstract we study non selfadjoint operator algebras that can be entirely understood via their finite dimensional representations in contrast with the elementary matricial description of finite dimensional c algebras in the non selfadjoint setting we show that an additional level of flexibility must be allowed motivated by this peculiarity we consider a natural non selfadjoint notion of residual finite dimensionality we identify sufficient conditions for the tensor algebra of a c correspondence to enjoy this property to clarify the connection with the usual self adjoint notion we investigate the residual finite dimensionality of the minimal and maximal c covers associated to an operator algebra

clebsch gordan coefficients and racah coefficients for the su 2 and su 1 1 groups as the discrete analogues of the poschl teller potential wavefunctions the clebsch gordan coefficients cgc and the racah coefficients for the su 2 and su 1 1 groups are studied as functions of a discrete variable it has been shown that cgc for su 2 and su 1 1 groups may be considered to be discrete analogues of wavefunctions for the one dimensional schrodinger equation with the poschl teller potential expressions for cgc and 6j symbols of the su 1 1 group have been found through the hahn and racah polynomials consideration is given to the asymptotic properties of eigenvalues and eigenfunctions of the hamiltonian of an asymmetric top and of the bargmann moshinsky operator omega

arveson extreme points span free spectrahedra let sm n mathbb r g denote g tuples of n times n real symmetric matrices given tuples x x 1 ldots x g in sm n 1 mathbb r g and y y 1 ldots y g in sm n 2 mathbb r g a matrix convex combination of x and y is a sum of the form begin aligned v 1 xv 1 v 2 y v 2 qquad qquad v 1 v 1 v 2 v 2 i n end aligned where v 1 mathbb r n rightarrow mathbb r n 1 and v 2 mathbb r n rightarrow mathbb r n 2 are contractions matrix convex sets are sets which are closed under matrix convex combinations a key feature of matrix convex combinations is that the g tuples x y and v 1 xv 1 v 2 y v 2 do not need to have the same size as a result matrix convex sets are a dimension free analog of convex sets while in the classical setting there is only one notion of an extreme point there are three main notions of extreme points for matrix convex sets ordinary matrix and absolute extreme points absolute extreme points are closely related to the classical arveson boundary a central goal in the theory of matrix convex sets is to determine if one of these types of extreme points for a matrix convex set minimally recovers the set through matrix convex combinations this article shows that every real compact matrix convex set which is defined by a linear matrix inequality is the matrix convex hull of its absolute extreme points and that the absolute extreme points are the minimal set with this property furthermore we give an algorithm which expresses a tuple as a matrix convex combination of absolute extreme points with optimal bounds similar results hold when working over the field of complex numbers rather than the reals

mehler heine type formulas for the krawtchouk polynomials abstract we derive mehler heine type asymptotic expansions for the krawtchouk polynomials k n x p n with n o n and x o n these formulas provide good approximations for k n x p n and determine the asymptotic limit of their zeros as n

tensor product of quotient hilbert modules abstract in this paper we present a unified approach to problems of tensor product of quotient modules of hilbert modules over c z and corresponding submodules of reproducing kernel hilbert modules over c z 1 z n and the doubly commutativity property of module multiplication operators by the coordinate functions more precisely for a reproducing kernel hilbert module h over c z 1 z n of analytic functions on the polydisc in c n which satisfies certain conditions we characterize the quotient modules q of h such that q is of the form q 1 q n for some one variable quotient modules q 1 q n for h the hardy module over polydisc h 2 d n this reduces to some recent results by izuchi nakazi and seto and the third author this is used to obtain a classification of co doubly commuting submodules for a class of reproducing kernel hilbert modules over the unit polydisc these results are applied to compute the cross commutators of co doubly commuting submodules moreover this provides further insight into the wandering subspaces and ranks of submodules of the hardy module our results include the case of weighted bergman modules over the unit polydisc in c n