invariant ergodic measures and the classification of crossed product c algebras abstract let g x be a minimal free continuous action of an infinite countable amenable group on an infinite compact metrizable space in this paper under the hypothesis that the invariant ergodic probability borel measure space e g x is compact and zero dimensional we show that the action has the small boundary property this partially answers an open problem in dynamical systems that asks whether a minimal free action of an amenable group has the small boundary property if its space m g x of invariant borel probability measures forms a bauer simplex in addition under the same hypothesis we show that dynamical comparison implies almost finiteness which was shown by kerr to imply that the crossed product is z stable finally we discuss some rank properties and provide two classifiability results for crossed products one of which is based on the work of elliott and niu

strongly self absorbing c algebras say that a separable unital c algebra v c is strongly self absorbing if there exists an isomorphism v such that lx ai e approximately unitarily equivalent homomorphisms we study this class of algebras which includes the cuntz algebras 2 ooo the uhf algebras of infinite type the jiang su algebra z and tensor products of 00 with uhf algebras of infinite type given a strongly self absorbing c algebra v we characterise when a separable c algebra absorbs v tensorially i e is p stable and prove closure properties for the class of separable testable c algebras finally we compute the possible if groups and prove a number of classification results which suggest that the examples listed above are the only strongly self absorbing c algebras

positive varieties of tree languages pin s variety theorem for positive varieties of string languages and varieties of finite ordered semigroups is proved for trees i e a bijective correspondence between positive varieties of tree languages and varieties of finite ordered algebras is established this in turn is extended to generalized varieties of finite ordered algebras which corresponds to steinby s generalized variety theorem also families of tree languages and classes of ordered algebras that are definable by ordered syntactic or translation monoids are characterized

simple c algebras with perforation abstract simple c algebras are constructed for which the ordered k 0 group is not weakly unperforated the algebras are approximately homogeneous and do not have slow dimension growth

characterizing ctl like logics on finite trees we associate a modal operator with each language belonging to a given class of regular tree languages and use the cascade product of tree automata to give an algebraic characterization of the expressive power of the resulting logic

decomposition rank and z stability we show that separable simple nonelementary unital c algebras with finite decomposition rank absorb the jiang su algebra mathcal z tensorially this has a number of consequences for elliott s program to classify nuclear c algebras by their k theory data in particular it completes the classification of c algebras associated to uniquely ergodic smooth minimal dynamical systems by their ordered k groups

non simple purely infinite c algebras the hausdorff case abstract local and global definitions of pure infiniteness for a c algebra a are compared and equivalence between them is obtained if the primitive ideal space of a is hausdorff and of finite dimension if a has real rank zero or if a is approximately divisible sufficient criteria are given for local pure infiniteness of tensor products they yield that exact simple tensorially non prime c algebras are purely infinite if they have no semi finite lower semi continuous trace one obtains that a is isomorphic to a o if a is 1 purely infinite separable stable nuclear and prim a is a hausdorff space not necessarily of finite dimension

the dixmier property and tracial states for c algebras a t was partially supported by an nserc postdoctoral fellowship and through the epsrc grant ep n00874x 1 acknowledgements we are grateful to luis santiago for helpful discussions at an early stage of this investigation we would also like to thank the referee for providing helpful comments which have led to a number of improvements

nuclear dimension and mathcal z stability simple separable unital monotracial and nuclear mathrm c algebras are shown to have finite nuclear dimension whenever they absorb the jiang su algebra mathcal z tensorially this completes the proof of the toms winter conjecture in the unique trace case

on topologically finite dimensional simple c algebras we show that if a simple c algebra a is topologically finite dimensional in a suitable sense then not only k0 a has certain good properties but a is even accessible to elliott s classification program more precisely we prove the following results

on a simple unital projectionless c algebra we construct a unital separable c algebra z as an analog of the hyperfinite type ii 1 factor besides being nuclear simple projectionless and infinite dimensional z has a unique tracial state and is kk equivalent to inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i the algebra of complex numbers it is shown that unital endomorphisms on z are approximately inner and that z is isomorphic to the infinite tensor product of its replicas it is also shown that a z a for certain interesting classes of unital simple nuclear c algebras a of real rank zero

nuclear dimension mathcal z stability and algebraic simplicity for stably projectionless c algebras the main result here is that a simple separable c algebra is mathcal z stable where mathcal z denotes the jiang su algebra if i it has finite nuclear dimension or ii it is approximately subhomogeneous with slow dimension growth this generalizes the main results of toms invent math 183 2 225 244 2011 and winter invent math 187 2 259 342 2012 to the nonunital setting as a consequence finite nuclear dimension implies mathcal z stability even in the case of a separable c algebra with finitely many ideals algebraic simplicity is established as a fruitful weakening of being simple and unital and the proof of the main result makes heavy use of this concept

the elliott conjecture for villadsen algebras of the first type abstract we study the class of simple c algebras introduced by villadsen in his pioneering work on perforated ordered k theory we establish six equivalent characterisations of the proper subclass which satisfies the strong form of elliott s classification conjecture two c algebraic z stability and approximate divisibility one k theoretic strict comparison of positive elements and three topological finite decomposition rank slow dimension growth and bounded dimension growth the equivalence of z stability and strict comparison constitutes a stably finite version of kirchberg s characterisation of purely infinite c algebras the other equivalences confirm for villadsen s algebras heretofore conjectural relationships between various notions of good behaviour for nuclear c algebras

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

on the classification of simple inductive limit c algebras ii the isomorphism theorem in this article it is proved that the invariant consisting of the scaled ordered k group and the space of tracial states together with the natural pairing between them is a complete invariant for the class of unital simple c algebras which can be expressed as the inductive limit of a sequence a 1 to a 2 to cdots to a n to cdots with a n bigoplus i 1 t n p n i m n i c x n i p n i where xn i is a compact metrizable space and pn i is a projection in m n i c xn i for each n and i and the spaces xn i are of uniformly bounded finite dimension note that the c algebras in the present class are not assumed to be of real rank zero as they were in eg2

covering dimension for nuclear c algebras we introduce the completely positive rank a notion of covering dimension for nuclear c algebras and analyze some of its properties the completely positive rank behaves nicely with respect to direct sums quotients ideals and inductive limits for abelian c algebras it coincides with covering dimension of the spectrum and there are similar results for continuous trace algebras as it turns out a c algebra is zero dimensional precisely if it is af we consider various examples particularly of one dimensional c algebras like the irrational rotation algebras the bunce deddens algebras or blackadar s simple unital projectionless c algebra finally we compare the completely positive rank to other concepts of noncommutative covering dimension such as stable or real rank

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

on the classification problem for nuclear c algebras we exhibit a counterexample to elliott s classification conjecture for simple separable and nuclear c algebras whose construction is elementary and demonstrate the necessity of extremely fine invariants in distinguishing both approximate unitary equivalence classes of automorphisms of such algebras and isomorphism classes of the algebras themselves the consequences for the program to classify nuclear c algebras are far reaching one has among other things that existing results on the classification of simple unital ah algebras via the elliott invariant of k theoretic data are the best possible and that these cannot be improved by the addition of continuous homotopy invariant functors to the elliott invariant

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

mathcal c _ 0 x algebras stability and strongly self absorbing mathcal c algebras we study permanence properties of the classes of stable and so called mathcal d stable mathcal c algebras respectively more precisely we show that a mathcal c 0 x algebra a is stable if all its fibres are provided that the underlying compact metrizable space x has finite covering dimension or that the cuntz semigroup of a is almost unperforated a condition which is automatically satisfied for mathcal c algebras absorbing the jiang su algebra mathcal z tensorially furthermore we prove that if mathcal d is a k 1 injective strongly self absorbing mathcal c algebra then a absorbs mathcal d tensorially if and only if all its fibres do again provided that x is finite dimensional this latter statement generalizes results of blanchard and kirchberg we also show that the condition on the dimension of x cannot be dropped along the way we obtain a useful characterization of when a mathcal c algebra with weakly unperforated cuntz semigroup is stable which allows us to show that stability passes to extensions of mathcal z absorbing mathcal c algebras

lifting kk elements asymptotic unitary equivalence and classification of simple c algebras let a and c be two unital simple c algebras with tracial rank zero suppose that c is amenable and satisfies the universal coefficient theorem denote by kke c a the set of those in kk c a for which k0 c 0 k0 a 0 and 1c 1a suppose that kke c a we show that there is a unital monomorphism c a such that suppose that c is a unital ah algebra and is a continuous affine map for which t p t p for all projections p in all matrix algebras of c and any tt a where t a is the simplex of tracial states of a and is the convex set of faithful tracial states of c we prove that there is a unital monomorphism c a such that induces both and suppose that h c a is a unital monomorphism and hom k1 c aff a we show that there exists a unital monomorphism c a such that h in kk c a t t h for all tracial states t and the associated rotation map can be given by denote by kkt c a the set of compatible pairs where kle c a and is a continuous affine map from t a to together with a result on asymptotic unitary equivalence in h lin asymptotic unitary equivalence and asymptotically inner automorphisms arxiv math 0703610 2007 this provides a bijection from the asymptotic unitary equivalence classes of unital monomorphisms from c to a to where is a subgroup related to vanishing rotation maps as an application combining these results with a result of w winter w winter localizing the elliott conjecture at strongly self absorbing c algebras arxiv 0708 0283v3 2007 we show that two unital amenable simple stable c algebras are isomorphic if they have the same elliott invariant and the tensor products of these c algebras with any uhf algebra have tracial rank zero in particular if a and b are two unital separable simple stable c algebras with unique tracial states which are inductive limits of c algebras of type i then they are isomorphic if and only if they have isomorphic elliott invariants

simple c algebras with locally finite decomposition rank abstract we introduce the notion of locally finite decomposition rank a structural property shared by many stably finite nuclear c algebras the concept is particularly relevant for elliott s program to classify nuclear c algebras by k theory data we study some of its properties and show that a simple unital c algebra which has locally finite decomposition rank real rank zero and which absorbs the jiang su algebra z tensorially has tracial rank zero in the sense of lin as a consequence of our result and of a theorem of elliott and gong any such c algebra if it additionally satisfies the universal coefficients theorem is approximately homogeneous of topological dimension at most 3 our result in particular confirms the elliott conjecture for the class of simple unital z stable ash algebras with real rank zero moreover it implies that simple unital z stable ah algebras with real rank zero not only have slow dimension growth in the ash sense but even in the ah sense

on the independence of k theory and stable rank for simple c algebras jiang and su and independently elliott discovered a simple nuclear infinite dimensional c algebra z having the same elliott invariant as the complex numbers for a nuclear c algebra a with weakly unperforated k group the elliott invariant of a z is isomorphic to that of a thus any simple nuclear c algebra a having a weakly unperforated k group which does not absorb z provides a counterexample to elliott s conjecture that the simple nuclear calgebras will be classified by the elliott invariant in the sequel we exhibit a separable infinite dimensional stably finite instance of such a non z absorbing algebra a and so provide a counterexample to the elliott conjecture for the class of simple nuclear infinite dimensional stably finite separable c algebras

covering dimension for nuclear c algebras ii the completely positive rank is an analogue of the topological covering dimension defined for nuclear c algebras via completely positive approximations these may be thought of as simplicial approximations of the algebra which leads to the concept of piecewise homogeneous maps and a notion of noncommutative simplicial complexes we introduce a technical variation of completely positive rank and show that the two theories coincide in many important cases furthermore we analyze some of their properties in particular we show that both theories behave nicely with respect to ideals and that they coincide with the covering dimension of the spectrum for certain continuous trace c algebras

the nuclear dimension of c algebras abstract we introduce the nuclear dimension of a c algebra this is a noncommutative version of topological covering dimension based on a modification of the earlier concept of decomposition rank our notion behaves well with respect to inductive limits tensor products hereditary subalgebras hence ideals quotients and even extensions it can be computed for many examples in particular it is finite for all uct kirchberg algebras in fact all classes of nuclear c algebras which have so far been successfully classified consist of examples with finite nuclear dimension and it turns out that finite nuclear dimension implies many properties relevant for the classification program surprisingly the concept is also linked to coarse geometry since for a discrete metric space of bounded geometry the nuclear dimension of the associated uniform roe algebra is dominated by the asymptotic dimension of the underlying space

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

on the stable rank of simple c algebras in 16 m a rieffel introduced the notion of stable rank for c algebras for a a unital c algebra the stable rank denoted by sr a is the least integer n such that the set of n tuples over a which generate a as a left ideal is dense in an if no such n exists we set sr a oo for unital c algebras the stable rank was shown to coincide with the bass stable rank which is defined for rings see 12 the stable rank can also be defined for nonunital c algebras but we shall only consider the unital case stable rank is related to nonstable k theory in 16 theorem 10 12 and 3 corollary 7 14 it was shown that if a is a simple unital c algebra of stable rank one then the natural map u a u a o 4 ki a is an isomorphism see also 17 2 and 18 it is a natural question which values occur as the stable rank of simple c algebras suppose that a is simple and infinite then by 7 proposition 1 5 a contains two isometries with orthogonal ranges and so by 16 proposition 6 5 sr a oo in the case of finite simple c algebras the following is known whenever a is simple and stably finite and b is a uhf algebra the tensor product a0 b has stable rank one see 19 this is also the case for every simple approximately homogeneous ah algebra with slow dimension growth see 4 and 8 so in short sr oo for every infinite simple c algebra and sr 1 for various classes of finite simple c algebras in this paper we answer the question above by showing that every n c n u oo is the stable rank of some simple finite c algebra more precisely it is shown that for every n 2 3 4 oo there exists a simple separable and unital ah algebra a of stable rank n in particular the group of invertible elements in a is not dense and like the examples in 20 a does not have slow dimension growth and is not approximately divisible see 5 i thank marius dadarlat for suggesting that i study ah algebras with connecting homomorphisms as described in the following section and juliana erlijman for drawing my attention to some of the powerful tools of differential topology and i thank george elliott for my stay at the fields institute where this paper was written

z stability and infinite tensor powers of c algebras we prove that under a mild hypothesis the infinite tensor power of a unital separable c algebra absorbs the jiang su algebra z tensorially combining this result with a recent theorem of winter we complete elliott s classification program for strongly self absorbing ash algebras we also give a succinct universal property for z in an ambient category so large that there are no unital separable c algebras without characters that are known to lie outside it this category contains the vast majority of our stock in trade separable amenable c algebras and is closed under passage to quotients and separable superalgebras in particular the category is closed under the formation of unital direct limits unital tensor products and crossed products by countable discrete groups finally we take a significant step toward the confirmation of elliott s classification conjecture for the c algebras of minimal diffeomorphisms

fundamental study tree algebras and varieties of tree languages we consider several aspects of wilke s t wilke an algebraic characterization of frontier testable tree languages theoret comput sci 154 1996 85 106 tree algebra formalism for representing binary labelled trees and compare it with approaches that represent trees as terms in the traditional way a convergent term rewriting system yields normal form representations of binary trees and contexts as well as a new completeness proof and a computational decision method for the axiomatization of tree algebras varieties of binary tree languages are compared with varieties of tree languages studied earlier in the literature we also prove a variety theorem thus solving a problem noted by several authors syntactic tree algebras are studied and compared with ordinary syntactic algebras the expressive power of the language of tree algebras is demonstrated by giving equational definitions for some well known varieties of binary tree languages

recognisability for algebras of infinite trees we develop an algebraic language theory for languages of infinite trees we define a class of algebras called w hyperclones and we show that a language of infinite trees is regular if and only if it is recognised by a finitary path continuous w hyperclone

an analogue of the thom isomorphism for crossed products of a c algebra by an action of r abstract in this paper we show the existence and uniqueness of a natural isomorphism oj of kj a with kj 1 a r j z 2 where a r is a c dynamical r system k is the functor of topological k theory and a r is the crossed product of a by the action of r the pimsner voiculescu exact sequence is obtained as a corollary we show that given an invariant trace on a with dual trace gt one has gto 1 u 1 2 i u u for any unitary u in the domain of the derivation of a associated to the action finally we show that the crossed product of c s3 continuous functions on the 3 sphere by a minimal diffeomorphism is a simple c algebra with no nontrivial idempotent

general varieties of tree languages we consider varieties of tree languages which are not restricted to a fixed ranked alphabet varieties of finite algebras that contain algebras of all finite types and a matching notion of varieties of congruences a variety theorem that yields isomorphisms between the lattices formed by these three types of varieties is proved to achieve this some basic universal algebra is suitably generalized and we define syntactic algebras so that two tree languages over any alphabets belong to the same varieties exactly in case their syntactic algebras are isomorphic many families of regular tree languages are shown to be varieties in our sense in particular we prove that every family of regular tree languages which can be characterized by syntactic monoids is such a variety but that the converse does not hold

homogenization of a generalized stefan problem in the context of ergodic algebras abstract we address the deterministic homogenization in the general context of ergodic algebras of a doubly nonlinear problem which generalizes the well known stefan model and includes the classical porous medium equation it may be represented by the differential inclusion for a real valued function u x t t u x e x u x x e x t u u f x e x t u on a bounded domain r n t 0 t together with initial boundary conditions where z x is strictly convex and z x t u is a c 1 convex function both with quadratic growth satisfying some additional technical hypotheses as functions of the oscillatory variable x u x t u and f x t u belong to the generalized besicovitch space b 2 associated with an arbitrary ergodic algebra a the periodic case was addressed by visintin 2007 based on the two scale convergence technique visintin s analysis for the periodic case relies heavily on the possibility of reducing two scale convergence to the usual l 2 convergence in the cartesian product r n where is the periodic cell this reduction is no longer possible in the case of a general ergodic algebra to overcome this difficulty we make essential use of the concept of two scale young measures for algebras with mean value associated with bounded sequences in l 2

classification of inductive limits of 1 dimensional nccw complexes abstract a classification result is obtained for the c algebras that are stably isomorphic to inductive limits of 1 dimensional noncommutative cw complexes with trivial k 1 group the classifying functor cu is defined in terms of the cuntz semigroup of the unitization of the algebra for the simple c algebras covered by the classification cu reduces to the ordered k 0 group the cone of traces and the pairing between them as an application of the classification it is shown that the crossed products by a quasi free action o 2 r are all isomorphic for a dense set of positive irrational numbers

algebraic recognizability of regular tree languages we propose a new algebraic framework to discuss and classify recognizable tree languages and to characterize interesting classes of such languages our algebraic tool called preclones encompasses the classical notion of syntactic algebra or minimal tree automaton but adds new expressivity to it the main result in this paper is a variety theorem a la eilenberg but we also discuss important examples of logically defined classes of recognizable tree languages whose characterization and decidability was established in recent papers by benedikt and segoufin and by bojanczyk and walukiewicz and can be naturally formulated in terms of pseudovarieties of preclones finally this paper constitutes the foundation for another paper by the same authors where first order definable tree languages receive an algebraic characterization

decomposition rank of subhomogeneous c algebras we analyze the decomposition rank a notion of covering dimension for nuclear c algebras introduced by e kirchberg and the author of subhomogeneous c algebras in particular we show that a subhomogeneous c algebra has decomposition rank n if and only if it is recursive subhomogeneous of topological dimension n and that n is determined by the primitive ideal space as an application we use recent results of q lin and n c phillips to show the following let a be the crossed product c algebra coming from a compact smooth manifold and a minimal diffeomorphism then the decomposition rank of a is dominated by the covering dimension of the underlying manifold

classifying crossed product c algebras i combine recent results in the structure theory of nuclear rm c algebras and in topological dynamics to classify certain types of crossed products in terms of their elliott invariants in particular transformation group rm c algebras associated to free minimal bbb z d actions on the cantor set with compact space of ergodic measures are classified by their ordered rm k theory in fact the respective statement holds for finite dimensional compact metrizable spaces provided that projections of the crossed products separate tracial states moreover rm c algebras associated to certain minimal homeomorphisms of spheres s 2n 1 are only determined by their spaces of invariant borel probability measures without a condition on the space of ergodic measures finally i show that for a large collection of classifiable rm c algebras crossed products by bbb z d actions are generically again classifiable