the cyclic and simplicial cohomology of the cuntz semigroup algebra abstract the main objective of this paper is to determine the simplicial and cyclic cohomology groups of the cuntz semigroup algebra l 1 s m we also determine the simplicial and cyclic cohomology of the tensor algebra of a banach space a class which includes the algebra on the free semigroup on m generators l 1 fs m in order to do so we first establish some general results which can be used when studying simplicial and cyclic cohomology of banach algebras in general we then turn our attention to l 1 s m showing that the cyclic cohomology groups of degree n vanish when n is odd and are one dimensional when n is even n 2 using the connes tzygan exact sequence these results are used to show that the simplicial cohomology groups of degree n vanish for n 1 a similar strategy is used for the tensor algebra of a banach space

residual finite dimensionality and representations of amenable operator algebras abstract we consider a version of a famous open problem formulated by kadison asking whether bounded representations of operator algebras are automatically completely bounded we investigate this question in the context of amenable operator algebras and we provide an affirmative answer for representations whose range is residually finite dimensional furthermore we show that weak closed amenable residually finite dimensional operator algebras are similar to c algebras and in particular have the property that all their bounded representations are completely bounded we prove our results for operator algebras having the so called total reduction property which is known to be weaker than amenability

coverings of k graphs k graphs are higher rank analogues of directed graphs which were first developed to provide combinatorial models for operator algebras of cuntz krieger type here we develop the theory of covering spaces for k graphs obtaining a satisfactory version of the usual topological classification in terms of subgroups of a fundamental group we then use this classification to describe the c algebras of covering k graphs as crossed products by coactions of homogeneous spaces generalizing recent results on the c algebras of graphs

uqosp 2 2 lattice models in this paper i construct lattice models with an underlying uqosp 2 2 superalgebra symmetry i find new solutions to the graded yang baxter equation these trigonometric r matrices depend on three continuous parameters the spectral parameter the deformation parameter q and the u 1 parameter b of the superalgebra it must be emphasized that the parameter q is generic and the parameter b does not correspond to the nilpotency parameter of the rational limits are given they also depend on the u 1 parameter and this dependence cannot be rescaled away i give the bethe ansatz solution of the lattice models built from some of these r matrices while for other matrices due to the particular nature of the representation theory of osp 2 2 i conjecture the result the parameter b appears as a continuous generalized spin finally i briefly discuss the problem of finding the ground state of these models

multi colour braid monoid algebras we define multi colour generalizations of braid monoid algebras and present explicit matrix representations which are related to two dimensional exactly solvable lattice models of statistical mechanics in particular we show that the two colour braid monoid algebra describes the yang baxter algebra of the critical dilute a d e models which were recently introduced by warnaar nienhuis and seaton 1992 as well as by roche 1992 these and other solvable models related to dense and dilute loop models are discussed in detail and it is shown that the solvability is a direct consequence of the algebraic structure it is conjectured that the baxterization of general multi colour braid monoid algebras will lead to the construction of further solvable lattice models

amenable and weakly amenable banach algebras with compact multiplication abstract we investigate amenable and weakly amenable banach algebras with compact multiplication any amenable banach algebra with compact multiplication is biprojective as a consequence every semisimple such algebra which has the approximation property is a topological direct sum of full matrix algebras in the radical case no such structure theorem is at hand we also investigate banach algebras which have a bounded approximate identity consisting of normalized powers of an element x any such banach algebra is either unital or radical if the algebra is also generated by x it is weakly amenable we construct a radical example with compact multiplication which moreover is an integral domain this furnishes a new example of a commutative weakly amenable non amenable radical banach algebra

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 spatial isomorphism problem for close separable nuclear c algebras the kadison kastler problem asks whether close c algebras on a hilbert space must be spatially isomorphic we establish this when one of the algebras is separable and nuclear we also apply our methods to the study of near inclusions of c algebras

when is the cuntz krieger algebra of a higher rank graph approximately finite dimensional evans d g and sims a when is the cuntz krieger algebra of a higher rank graph approximately finite dimensional preprint arxiv 1112 4549v1

relative yoneda cohomology for operator spaces abstract we provide two alternate presentations of the completely bounded hochschild cohomology one as a relative yoneda cohomology i e as equivalence classes of n resolutions which are relatively split and the second as a derived functor the first presentation makes clear the importance of certain relative notions of injectivity projectivity and amenability which we introduce and study we prove that every von neumann algebra is relatively injective as a bimodule over itself and consequently h n cb m m 0 for any von neumann algebra m a result obtained earlier by christensen and sinclair we prove that the relatively amenable c algebras are precisely the nuclear c algebras and hence exactly those which are amenable as banach algebras in a similar vein we prove that the only relatively projective c algebras are finite dimensional this result implies that the only c algebras that are projective as banach algebras are finite dimensional a result first obtained by selivanov and helemskii in a slightly different direction we prove that b h viewed as a bimodule over the disk algebra with the left action given by multiplication by a coisometry and the right action given by multiplication by an isometry is an injective module this result is in some sense a generalization of the sz nagy foias commutant lifting theorem or of the hypoprojectivity introduced by douglas and the author

weak amenability of module extensions of banach algebras we start by discussing general necessary and sufficient conditions for a module extension banach algebra to be n weakly amenable for n 0 1 2 then we investigate various special cases all these case studies finally provide us with a way to construct an example of a weakly amenable banach algebra which is not 3 weakly amenable this answers an open question raised by h g dales f ghahramani and n gronbaek

systemes dynamiques non commutatifs et moyennabilite on introduit la notion de fonction de type positif relativement a un systeme dynamique c dont les proprietes sont analogues a celles des fonctions de type positif classiques on montre que dans le cas d un groupe discret g agissant sur une algebre de von neumann m la moyennabilite de l action est equivalente a l existence d une suite bornee de fonctions de type positif sur g a support fini a valeurs dans le centre z m de m qui converge vers 1 pour la topologie de la convergence simple ultrafaible

product systems over right angled artin semigroups we build upon mac lane s definition of a tensor category to introduce the concept of a product system that takes values in a tensor groupoid g we show that the existing notions of product systems fit into our categorical framework as do the k graphs of kumjian and pask we then specialize to product systems over right angled artin semigroups these are semigroups that interpolate between free semigroups and free abelian semigroups for such a semigroup we characterize all product systems which take values in a given tensor groupoid g in particular we obtain necessary and sufficient conditions under which a collection of k 1 graphs form the coordinate graphs of a k graph

the cyclic and simplicial cohomology of l 1 mathbf n let a l 1 z be the unital semigroup algebra of n we show that the cyclic cohomology groups hc n a a vanish when n is odd and are one dimensional when n is even n 2 using connes exact sequence these results are used to show that the simplicial cohomology groups h n a a vanish for n 2 the results obtained are extended to unital algebras l 1 s for some other semigroups of r

higher dimensional generalizations of the thompson groups abstract motivated by the goal of generalizing the cuntz krieger algebras and making heavy use of the pioneering work of robertson and steger kumjian and pask generalized the notion of a directed graph to what they termed a higher rank graph the c algebras constructed from such higher rank graphs have proved to be highly interesting higher rank graphs are in fact a class of cancellative categories and so the one vertex higher rank graphs are therefore a class of cancellative monoids in this paper we call them k monoids finite direct products of free monoids belong to this family but there are many examples not of this form it is well known that the classical thompson groups arise from the free monoids and it is easy to show that brin s higher dimensional thompson groups arise from finite direct products of free monoids in this paper we generalize these constructions and show how to construct a family of new groups with simple commutator subgroups from arbitrary aperiodic k monoids thereby subsuming all the above examples although we give simple direct constructions of our groups a key feature of our paper is that we also show that they arise as topological full groups of the usual etale groupoids associated with k monoids in this way our groups are naturally associated with c algebras the question of the mutual isomorphisms amongst this family of groups appears to be a delicate one and is not handled here

on the k theory of twisted higher rank graph c algebras abstract we investigate the k theory of twisted higher rank graph algebras by adapting parts of elliott s computation of the k theory of the rotation algebras we show that each 2 cocycle on a higher rank graph taking values in an abelian group determines a continuous bundle of twisted higher rank graph algebras over the dual group we use this to show that for a circle valued 2 cocycle on a higher rank graph obtained by exponentiating a real valued cocycle the k theory of the twisted higher rank graph algebra coincides with that of the untwisted one

z stability of crossed products by strongly outer actions ii we consider a crossed product of a unital simple separable nuclear stably finite z stable c algebra a by a strongly outer cocycle action of a discrete countable amenable group under the assumption that a has finitely many extremal tracial states and is elementary amenable we show that the twisted crossed product c algebra is z stable as an application we also prove that all strongly outer cocycle actions of the klein bottle group on z are cocycle conjugate to each other this is the first classification result for actions of non abelian infinite groups on stably finite c algebras

eigenvalues of casimir invariants for unitary irreps of a fully explicit formula for the eigenvalues of casimir invariants for u q gl m n is given which applies to all unitary irreps this is achieved by making some interesting observations on atypicality indices for irreps occurring in the tensor product of unitary irreps of the same type these results have applications in the determination of link polynomials arising from unitary irreps of u q gl m n

separable representations kms states and wavelets for higher rank graphs letbe a strongly connected finite higher rank graph in this paper we construct faithful representations of c on certain separable hilbert spaces of the form l 2 x by introducing the notion of a semibranching function system a generalization of the semibranching function systems studied by marcolli and paolucci in particular we obtain a faithful representation of c on l 2 1 m where m is the perron frobenius probability measure on the infinite path space 1 recently studied by an huef laca raeburn and sims we also show how a semibranching function system gives rise to kms states for c for the higher rank graphs of robertson and steger we also obtain a faithful representation of c on l 2 x where x is a fractal subspace of 0 1 by embedding 1 into 0 1 as a fractal subset x of 0 1 in this latter case we additionally show that there exists a kms state for c whose inverse temperature is equal to the hausdorff dimension of x finally we construct a wavelet system for l 2 1 m by generalizing the work of marcolli and paolucci from graphs to higher rank graphs

twisted quantum affine superalgebra invariant r matrices and a new integrable electronic model we describe the twisted affine superalgebra sl 2 2 2 and its quantized version u q sl 2 2 2 we investigate the tensor product representation of the four dimensional grade star representation for the fixed point sub superalgebra u q osp 2 2 we work out the tensor product decomposition explicitly and find that the decomposition is not completely reducible associated with this four dimensional grade star representation we derive two u q osp 2 2 invariant r matrices one of them corresponds to u q sl 2 2 2 and the other to u q osp 2 2 1 using the r matrix for u q sl 2 2 2 we construct a new u q osp 2 2 invariant strongly correlated electronic model which is integrable in one dimension interestingly this model reduces in the q 1 limit to the one proposed by essler et al which has a larger sl 2 2 symmetry

rohlin flows on the cuntz algebra o it is shown that certain quasi free flows on the cuntz algebra o have the rohlin property and therefore are cocycle conjugate with each other this in particular shows that any unital separable nuclear purely infinite simple c algebra has a rohlin flow

amenability for twisted covariance algebras and group c algebras abstract the basic question studied in this paper is when is a twisted covariance c algebra c g a f amenable this includes as a special case the question when is a group c algebra c g amenable we show in particular that c h a f is amenable if c g a f is amenable and h is a closed normal subgroup of g other results that we prove use the derivation and injective characterizations for amenable c algebras

affine buildings tiling systems and higher rank cuntz krieger algebras to an r dimensional subshift of finite type satisfying certain special properties we associate a c algebra a this algebra is a higher rank version of a cuntz krieger algebra in particular it is simple purely infinite and nuclear we study an example if is a group acting freely on the vertices of an a2 building with finitely many orbits and if is the boundary of that building then c o is the algebra associated to a certain two dimensional subshift

solutions of the quantum yang baxter equation with extra nonadditive parameters we present a systematic technique to construct solutions to the yang baxter equation which depend not only on a spectral parameter but in addition on further continuous parameters these extra parameters enter the yang baxter equation in a similar way to the spectral parameter but in a non additive form we exploit the fact that quantum non compact algebras such as uq su 1 1 and type i quantum superalgebras such as uq gl 1 mod 1 and uq gl 2 mod 1 are known to admit non trivial one parameter families of infinite dimensional and finite dimensional irreps respectively even for generic q we develop a technique for constructing the corresponding spectral dependent r matrices as examples we work out the the r matrices for the three quantum algebras mentioned above in certain representations

semigroupoid c algebras abstract a semigroupoid is a set equipped with a partially defined associative operation given a semigroupoid we construct a c algebra o from it we then present two main examples of semigroupoids namely the markov semigroupoid associated to an infinite 0 1 matrix and the semigroupoid associated to a row finite higher rank graph without sources in both cases the semigroupoid c algebra is shown to be isomorphic to the algebras usually attached to the corresponding combinatorial object namely the cuntz krieger algebras and the higher rank graph c algebras respectively in the case of a higher rank graph d it follows that the dimension function d is superfluous for defining the corresponding c algebra

generalized notions of amenability ii this paper continues the investigation of the first two authors begun in part i it is shown that approximate amenability and approximate contractibility are the same properties as are uniform approximate amenability and amenability bounded approximate contractibility and bounded approximate amenability are characterized by the existence of suitable operator bounded approximate identities for the diagonal ideal results are given on banach sequence algebras lipschitz algebras and beurling algebras and on the crucial role of approximate identities a new proof is given for a result due to n gronbaek on characterizing of amenability for beurling algebras

perturbations of nuclear c algebras kadison and kastler introduced a natural metric on the collection of all c subalgebras of the bounded operators on a separable hilbert space they conjectured that sufficiently close algebras are unitarily conjugate we establish this conjecture when one algebra is separable and nuclear we also consider one sided versions of these notions and we obtain embeddings from certain near inclusions involving separable nuclear c algebras at the end of the paper we demonstrate how our methods lead to improved characterisations of some of the types of algebras that are of current interest in the classification programme

boundary c algebras of triangle geometries abstract let be a building of type a 2 and order q with maximal boundary let be a group of type preserving automorphisms of which acts regularly on the chambers of then the crossed product c algebra c is isomorphic to m 3 q 1 o q 2 o q 2 where o n denotes the cuntz algebra generated by n isometries whose range projections sum to the identity operator

the interplay between k graphs and the yang baxter equation in this paper we initiate the study of the interplay between k graphs and the yang baxter equation for this we provide two very different perspectives on one hand we show that the set of all set theoretic solutions of the yang baxter equation is a special class of single vertex k graphs as a consequence we construct an infinite family of large solutions of the yang baxter equation from an arbitrarily given one on the other hand we prove that all single vertex k graphs are yb semigroups of square free involutive solutions of the yang baxter equation other various connections are also investigated

some reformulated properties of jacobian elliptic functions abstract various properties of jacobian elliptic functions can be put in a form that remains valid under permutation of the first three of the letters c d n and s that are used in pairs to name the functions in most cases 12 formulas are thereby replaced by three one for the three names that end in s one for the three that begin with s and one for the six that do not involve s the properties thus unified in the present paper are linear relations between squared functions 16 relations being replaced by five differential equations and indefinite integrals of odd powers of a single function in the last case the unification entails the elementary function r c x y r f x y y where r f x y z is the symmetric elliptic integral of the first kind explicit expressions in terms of r c are given for integrals of first and third powers and alternative expressions are given with r c replaced by inverse circular inverse hyperbolic or logarithmic functions three recurrence relations for integrals of odd powers hold also for integrals of even powers

table of integrals of squared jacobian elliptic functions and reductions of related hypergeometric r functions any product of real powers of jacobian elliptic functions can be written in the form cs m1 u k ds m2 u k ns m3 u k if all three m s are even integers the indefinite integral of this product with respect to u is a constant times a multivariate hypergeometric function r a b 1 b 2 b 3 x y z with half odd integral b s and a b 1 b 2 b 3 1 showing it to be an incomplete elliptic integral of the second kind unless all three m s are 0 permutations of c d and n in the integrand produce the same permutations of the variables x y z cs 2 ds 2 ns 2 allowing as many as six integrals to take a unified form thirty r functions of the type specified incorporating 136 integrals are reduced to a new choice of standard elliptic integrals obtained by permuting x y and z in r d x y z r 3 2 1 2 1 2 3 2 x y z which is symmetric in its first two variables and has an efficient algorithm for numerical computation

a note on graded yang baxter solutions as braid monoid invariants we construct two osp n mod 2m solutions of the graded yang baxter equation by using the algebraic braid monoid approach the factorizable s matrix interpretation of these solutions is also discussed

the uniqueness of cuntz krieger type algebras we introduce a class of c algebras which can be viewed as a generalization of the classical cuntz krieger algebras the feature is that it is a flexible generators and relations concept the main result is a canonical uniqueness theorem stating that the c algebras of this class are uniquely determined by their generators and relations we can show that rank one cuntz krieger algebras with infinitely large transition matrices fall in this class and this provides an alternative proof of exel and laca further we elaborate a subclass of rank two cuntz krieger algebras inspired by shifts of finite type in dimension two with an infinite set of generators and relations

exactly solvable quantum spin ladders associated with the orthogonal and symplectic lie algebras we extend the results of spin ladder models associated with the lie algebras su 2 n to the case of the orthogonal and symplectic algebras o 2 n sp 2 n where n is the number of legs for the system two classes of models are found whose symmetry either orthogonal or symplectic has an explicit n dependence integrability of these models is shown for an arbitrary coupling of xx type rung interactions and applied magnetic field term

primitive ideal space of higher rank graph c algebras and decomposability abstract in this paper we describe primitive ideal space of the c algebra c associated to any locally convex row finite k graph to do this we will apply the farthing s desourcifying method on a recent result of carlsen kang shotwell and sims we also characterize certain maximal ideals of c furthermore we study the decomposability of c we apply the description of primitive ideals to show that if i is a direct summand of c then it is gauge invariant and isomorphic to a certain k graph c algebra so we may characterize decomposable higher rank c algebras by giving necessary and sufficient conditions for the underlying k graphs moreover we determine all such c algebras which can be decomposed into a direct sum of finitely many indecomposable c algebras

kms states on c algebras associated to higher rank graphs consider a higher rank graph of rank k both the cuntz krieger algebra and the toeplitz cuntz krieger algebra of the graph carry natural gauge actions of the torus tk and restricting these gauge actions to one parameter subgroups of tk gives dynamical systems involving actions of the real line we study the kms states of these dynamical systems we find that for large inverse temperatures the simplex of kms states on the toeplitz cuntz krieger algebra has dimension d one less than the number of vertices in the graph we also show that there is a preferred dynamics for which there is a critical inverse temperature c for larger than c there is a d dimensional simplex of kms states when c and the one parameter subgroup is dense there is a unique kms state and this state factors through the cuntz krieger algebra as in previous studies for k 1 our main tool is the perron frobenius theory for irreducible non negative matrices though here we need a version of the theory for commuting families of matrices

symmetry in c d n of jacobian elliptic functions abstract the relation connecting the symmetric elliptic integral r f with the jacobian elliptic functions is symmetric in the first three of the four letters c d n and s that are used in ordered pairs to name the 12 functions a symbol p q ps 2 u k qs 2 u k p q c d n is independent of u and allows formulas for differentiation bisection duplication and addition to remain valid when c d and n are permuted the five transformations of first order which change the argument and modulus of the functions take a unified form in which they correspond to the five nontrivial permutations of c d and n there are 18 transformations of second order including landen s and gauss s transformations comprising three sets of six the sets are related by permutations of the original functions cs ds and ns and there are only three sets because each set is symmetric in two of these the six second order transformations in each set are related by first order transformations of the transformed functions and all 18 take a unified form all results are derived from properties of r f without invoking weierstrass functions or theta functions

simple purely infinite c algebras and n filling actions let n be a positive integer we introduce a concept which we call the n filling property for an action of a group on a separable unital c algebra a if a c is a commutative unital c algebra and the action is induced by a group of homeomorphisms of then the n filling property reduces to a weak version of hyperbolicity the n filling property is used to prove that certain crossed product c algebras are purely infinite and simple a variety of group actions on boundaries of symmetric spaces and buildings have the n filling property an explicit example is the action of sln z on the projective n space

algebraic connections universal bimodules and entire cyclic cohomology a new method of constructing homotopies suitable for entire cyclic cohomology is presented as a result the periodic and entire cyclic cohomology of banach algebras of finite cohomological dimension are shown to be isomorphic the same method can be used to calculate the algebraic entire cyclic cohomology of non commutative tori

a fixed point theorem for l1 spaces we prove a fixed point theorem for a family of banach spaces notably for l 1 applications include the optimal answer to the derivation problem for group algebras which originated in the 1960s

amenability and uniqueness abstract the main result of this paper is a characterization of properly infinite injective von neumann algebras and of nuclear c algebras by using a uniqueness theorem based on generalizations of voiculescu s famous weyl von neumann theorem

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

cyclic cohomology of banach algebras of quivers and their inverse semigroups abstract in this paper we consider the path semigroup l 1 algebra for a quiver and the inverse semigroup l 1 algebra of a quiver the latter of which can be used in the construction of cuntz krieger algebras the main objectives of the paper are to determine the simplicial and cyclic cohomology groups of these algebras first we determine the simplicial and cyclic cohomology of the path algebra of the quiver showing the simplicial cohomology groups of dimension n vanish for n 1 we then determine the simplicial and cyclic cohomology of the inverse semigroup algebra the work uses the connes tzygan long exact sequence

type iii representations and automorphisms of some separable nuclear c algebras abstract powers proved decades ago that if two cyclic representations 1 and 2 of a uhf algebra a satisfy that 1 a 2 a m there is an automorphism of a such that 1 and 2 are quasi equivalent this was recently extended to the class of simple separable c algebras when m is of type i in this paper we extend this result to some class of simple separable nuclear c algebras when m is of type iii in particular we show that the above result obtained by powers for the uhf algebras also holds for the class of purely infinite simple separable c algebras classified by kirchberg and phillips and for the class of approximately homogeneous simple separable unital c algebra with unique tracial state

c algebras associated with hilbert c quad modules of c textile dynamical systems abstract a c textile dynamical system a consists of a unital c algebra a two families of endomorphisms and a a of a and certain commutation relations among them it yields a two dimensional subshift and a multistructure hilbert c bimodule which we call a hilbert c quad module we introduce a c algebra from the hilbert c quad module as a two dimensional analogue of pimsner s construction of c algebras from hilbert c bimodules we study the c algebras defined by the hilbert c quad modules and prove that they have universal properties subject to certain operator relations we also present examples arising from commuting matrices

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

amenability and virtual diagonals for von neumann algebras abstract it is shown that a certain universal bimodule over a von neumann algebra is automatically normal this leads to a simple direct proof of haagerup s result that a von neumann algebra is amenable or equivalently injective if and only if it has a normal virtual diagonal haagerup has used this fact to prove that nuclear c algebras are amenable

strict comparison and z absorption of nuclear c algebras for any unital separable simple infinite dimensional nuclear c algebra with finitely many extremal traces we prove that mathcal z absorption strict comparison and property si are equivalent we also show that any unital separable simple nuclear c algebra with tracial rank zero is approximately divisible and hence is mathcal z absorbing

generalised morphisms of k graphs k morphs in a number of recent papers k l graphs have been constructed from k graphs by inserting new edges in the last l dimensions these constructions have been motivated by c algebraic considerations so they have not been treated systematically at the level of higher rank graphs themselves here we introduce k morphs which provide a systematic unifying framework for these various constructions we think of k morphs as the analogue at the level of k graphs of c correspondences between c algebras to make this analogy explicit we introduce a category whose objects are k graphs and whose morphisms are isomorphism classes of k morphs we show how to extend the assignment c to a functor from this category to the category whose objects are c algebras and whose morphisms are isomorphism classes of c correspondences

truncated global transfer matrix trigonometric goryachev chaplygin gyrostat the rtt relation with truncation is solved for the r matrix associated with the six vertex model it turns out that the finite expansion of the global solution of the transfer matrix t u in terms of the spectral parameter u can be realized by the defined algebra formed by and the non commutative coordinates the hamiltonian of the system has been found which is the trigonometric deformation of the goryachev chaplygin gyrostat

yang baxterization of braid group representations for a given braid group representation bgr a process of the yang baxterization is formulated to generate solutions of the yang baxter equation ybe when a bgr admits the birman wenzl bw algebraic structure this process can be explicitly passed through and two types of trigonometric solutions of ybe are generated from such a bgr these two solutions have the essential difference to each other and both of them preserve the crossing symmetry property if the given bgr has by taking certain reduction on the bw algebra the rational solution is also generated a practical condition to judge whether a bgr satisfies the bw algebra is given from which one finds that not only the familiar bgrs of 5 7 9 but also some new ones obtained recently in 12 have the bw structure thus they can be explicitly yang baxterized to solutions of the ybe

higher dimensional virtual diagonals and ideal cohomology for triangular algebras we investigate the cohomology of non self adjoint algebras using virtual diagonals and their higher dimensional generalizations we show that infinite dimensional nest algebras always have non zero second cohomology by showing that they cannot possess 2 virtual diagonals in the case of the upper triangular atomic nest algebra we exhibit concrete modules for non vanishing cohomology

on the weak amenability of a x and its relation with the approximation property abstract we investigate the weak amenability of the algebra a x of approximable operators on a banach space x and its relation with the bounded approximation property in particular it will be shown that the bounded approximation property is neither necessary nor sufficient for the weak amenability of a x

the primitive ideals of the cuntz krieger algebra of a row finite higher rank graph with no sources abstract we catalogue the primitive ideals of the cuntz krieger algebra of a row finite higher rank graph with no sources each maximal tail in the vertex set has an abelian periodicity group of finite rank at most that of the graph the primitive ideals in the cuntz krieger algebra are indexed by pairs consisting of a maximal tail and a character of its periodicity group the cuntz krieger algebra is primitive if and only if the whole vertex set is a maximal tail and the graph is aperiodic

cohen host type idempotent theorems for representations on banach spaces and applications to figa talamanca herz algebras abstract let g be a locally compact group and let r g denote the ring of subsets of g generated by the left cosets of open subsets of g the cohen host idempotent theorem asserts that a set lies in r g if and only if its indicator function is a coefficient function of a unitary representation of g on some hilbert space we prove related results for representations of g on certain banach spaces we apply our cohen host type theorems to the study of the figa talamanca herz algebras a p g with p 1 for arbitrary g we characterize those closed ideals of a p g that have an approximate identity bounded by 1 in terms of their hulls furthermore we characterize those g such that a p g is 1 amenable for some and equivalently for all p 1 these are precisely the abelian groups

bethe subalgebras in affine birman murakami wenzl algebras and flat connections for q kz equations commutative sets of jucys murphy elements for affine braid groups of types were defined construction of r matrix representations of the affine braid group of type and its distinguished commutative subgroup generated by the type jucys murphy elements are given we describe a general method to produce flat connections for the two boundary quantum knizhnik zamolodchikov equations as necessary conditions for sklyanin s type transfer matrix associated with the two boundary multicomponent zamolodchikov algebra to be invariant under the action of the type jucys murphy elements we specify our general construction to the case of the birman murakami wenzl algebras bmw algebras for short as an application we suggest a baxterization of the dunkl cherednik elements in the double affine hecke algebra of type a

the yang baxter equation and invariants of links on considere le fait que l on peut construire des invariants d isotopie p et f pour des liens en utilisant certaines solutions des equations de yang baxter

on twisted higher rank graph c algebras we define the categorical cohomology of a k graphand show that the first three terms in this cohomology are isomorphic to the corresponding terms in the cohomology defined in our previous paper this leads to an alternative characterisation of the twisted k graph c algebras introduced there we prove a gauge invariant uniqueness theorem and use it to show that every twisted k graph c algebra is isomorphic to a twisted groupoid c algebra we deduce criteria for simplicity prove a cuntz krieger uniqueness theorem and establish that all twisted k graph c algebras are nuclear and belong to the bootstrap class

dilute birman wenzl murakami algebra and dn 1 2 models a dilute generalization of the birman wenzl murakami algebra is considered it can be baxterized to a solution of the yang baxter algebra the dn 1 2 vertex models constitute a series of solvable lattice models which realize this algebraic structure they can be regarded as a dilute version of the bn 1 vertex models

a generalized cuntz krieger uniqueness theorem for higher rank graphs we present a uniqueness theorem for k graph c algebras that requires neither an aperiodicity nor a gauge invariance assumption specifically we prove that for the injectivity of a representation of a k graph c algebra it is sufficient that the representation be injective on a distinguished abelian c subalgebra a crucial part of the proof is the application of an abstract uniqueness theorem which says that such a uniqueness property follows from the existence of a jointly faithful collection of states on the ambient c algebra each of which is the unique extension of a state on the distinguished abelian c subalgebra

atomic representations of rank 2 graph algebras abstract we provide a detailed analysis of atomic representations of rank 2 graphs on a single vertex they are completely classified up to unitary equivalence and decomposed into a direct sum or direct integral of irreducible atomic representations the building blocks are described as the minimal dilations of defect free representations modelled on finite groups of rank 2

a new braid like algebra for baxterisation we introduce a new baxterisation for r matrices that depend separately on two spectral parameters the baxterisation is based on a new algebra close to but different from the braid group we study representations of this new algebra on the vector space mathbb c m otimes n when the generators act locally the ones for m 2 are completely classified we also introduce some representations for generic m they allow us to recover the r matrix of the multi species generalization of the totally asymmetric simple exclusion process with different hopping rates

homology for higher rank graphs and twisted c algebras abstract we introduce a homology theory for k graphs and explore its fundamental properties we establish connections with algebraic topology by showing that the homology of a k graph coincides with the homology of its topological realisation as described by kaliszewski et al we exhibit combinatorial versions of a number of standard topological constructions and show that they are compatible from a homological point of view with their topological counterparts we show how to twist the c algebra of a k graph by a t valued 2 cocycle and demonstrate that examples include all noncommutative tori in the appendices we construct a cubical set q from a k graph and demonstrate that the homology and topological realisation of coincide with those of q as defined by grandis

equivalent isotropic osp 2 2 spin chains and its bethe ansatz analysis in this paper we discuss the equivalence between two integrable and isotropic osp 2 2 chains via the braid monoid approach we use the analytical bethe ansatz approach in order to solve the spectrum of these systems the ground state is parametrized by a special structure of the bethe ansatz roots their interpretation in terms of correlated fermionic models is briefly discussed

weak amenability of triangular banach algebras let a and b be unital banach algebras and let m be a banach a b module then t a m 0 b becomes a triangular banach algebra when equipped with the banach space norm a m 0 b a a m m b b a banach algebra t is said to be n weakly amenable if all derivations from t into its n th dual space t n are inner in this paper we investigate arens regularity and n weak amenability of a triangular banach algebra t in relation to that of the algebras a b and their action on the module m