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

quasidiagonal morphisms and homotopy abstract adapting certain methods of voiculescu we show that the quasidiagonality of c algebras is invariant under completely positive deformations coming from asymptotic morphisms we define a notion of quasidiagonal morphism the quasidiagonality of morphisms from exact separable c algebras is shown to be invariant under completely positive deformations

tracial state space with non compact extreme boundary let a be a unital simple separable c algebra if a is nuclear and infinite dimensional it is known that strict comparison of positive elements is equivalent to z stability if the extreme boundary of its tracial state space is compact and of finite covering dimension here we provide the first proof of this result in the case of certain non compact extreme boundaries additionally if a has strict comparison of positive elements it is known that the cuntz semigroup of this c algebra is recovered functorially from the murray von neumann semigroup and the tracial state space whenever the extreme boundary of the tracial state space is compact and of finite covering dimension we extend this result to the case of a countable extreme boundary with finitely many cluster points

c stability of discrete groups abstract a group may be considered c stable if almost representations of the group in a c algebra are always close to actual representations we initiate a systematic study of which discrete groups are c stable or only stable with respect to some subclass of c algebras e g finite dimensional c algebras we provide criteria and invariants for stability of groups and this allows us to completely determine stability non stability of crystallographic groups surface groups virtually free groups and certain baumslag solitar groups we also show that among the non trivial finitely generated torsion free 2 step nilpotent groups the only c stable group is z

abelian mathrm c ast subalgebras of mathrm c ast algebras of real rank zero and inductive limit mathrm c ast algebras

classification of c homomorphisms from c0 0 1 to a c algebra abstract a class of c algebras is described for which the c homomorphisms from c 0 0 1 to the algebra may be classified by means of the cuntz semigroup functor examples are given of algebras simple and non simple for which this classification fails it is shown that a suitable suspension of the cuntz semigroup functor deals successfully with some of these counterexamples

cartan subalgebras in c algebras existence and uniqueness we initiate the study of cartan subalgebras in c algebras with a particular focus on existence and uniqueness questions for homogeneous c algebras these questions can be analyzed systematically using the theory of fiber bundles for group c algebras while we are able to find cartan subalgebras in c algebras of many connected lie groups there are classes of discrete groups for instance non abelian free groups whose reduced group c algebras do not have any cartan subalgebras moreover we show that uniqueness of cartan subalgebras usually fails for classifiable c algebras however distinguished cartan subalgebras exist in some cases for instance in nuclear uniform roe algebras

amenability and paradoxicality in semigroups and c algebras abstract we analyze the dichotomy amenable paradoxical in the context of discrete countable unital semigroups and corresponding semigroup rings we consider also folner type characterizations of amenability and give an example of a semigroup whose semigroup ring is algebraically amenable but has no folner sequence in the context of inverse semigroups s we give a characterization of invariant measures on s in the sense of day in terms of two notions domain measurability and localization given a unital representation of s in terms of partial bijections on some set x we define a natural generalization of the uniform roe algebra of a group which we denote by r x we show that the following notions are then equivalent 1 x is domain measurable 2 x is not paradoxical 3 x satisfies the domain folner condition 4 there is an algebraically amenable dense subalgebra of r x 5 r x has an amenable trace 6 r x is not properly infinite and 7 0 1 in the k 0 group of r x we also show that any tracial state on r x is amenable moreover taking into account the localization condition we give several c algebraic characterizations of the amenability of x finally we show that for a certain class of inverse semigroups the quasidiagonality of c r x implies the amenability of x the reverse implication which is a natural generalization of rosenberg s conjecture to this context is false

computing bf l _ 2 gain of finite horizon systems with boundary conditions a bisection algorithm is developed for computing the l2 gain of a finite horizon system with boundary conditions upper and lower bounds of the gain are also derived for the initial step of the algorithm

on groups with quasidiagonal c algebras abstract we examine the question of quasidiagonality for c algebras of discrete amenable groups from a variety of angles we give a quantitative version of rosenberg s theorem via paradoxical decompositions and a characterization of quasidiagonality for group c algebras in terms of embeddability of the groups we consider several notable examples of groups such as topological full groups associated with cantor minimal systems and abels celebrated example of a finitely presented solvable group that is not residually finite and show that they have quasidiagonal c algebras finally we study strong quasidiagonality for group c algebras exhibiting classes of amenable groups with and without strongly quasidiagonal c algebras

strongly self absorbing c dynamical systems iii abstract in this paper we accomplish two objectives firstly we extend and improve some results in the theory of semi strongly self absorbing c dynamical systems which was introduced and studied in previous work in particular this concerns the theory when restricted to the case where all the semi strongly self absorbing actions are assumed to be unitarily regular which is a mild technical condition the central result in the first part is a strengthened version of the equivariant mcduff type theorem where equivariant tensorial absorption can be achieved with respect to so called very strong cocycle conjugacy secondly we establish completely new results within the theory this mainly concerns how equivariantly z stable absorption can be reduced to equivariantly uhf stable absorption with respect to a given semi strongly self absorbing action combining these abstract results with known uniqueness theorems due to matui and izumi matui we obtain the following main result if g is a torsion free abelian group and d is one of the known strongly self absorbing c algebras then strongly outer g actions on d are unique up to very strong cocycle conjugacy this is new even for z 3 actions on the jiang su algebra

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

continuous fields of kirchberg c algebras abstract in this paper we study the c algebras associated to continuous fields over locally compact metrizable zero dimensional spaces whose fibers are kirchberg c algebras satisfying the uct we show that these algebras are inductive limits of finite direct sums of kirchberg algebras and they are classified up to isomorphism by topological invariants

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

unitary orbits of self adjoint operators in simple z stable c algebras abstract we prove that in a simple unital exact z stable c algebra the distance between the unitary orbits of self adjoint elements with connected spectrum is completely determined by spectral data this fails without the assumption of z stability

a functional link based neurofuzzy network for nonlinear system control this study presents a functional link based neurofuzzy network flnfn structure for nonlinear system control the proposed flnfn model uses a functional link neural network flnn to the consequent part of the fuzzy rules this study uses orthogonal polynomials and linearly independent functions in a functional expansion of the flnn thus the consequent part of the proposed flnfn model is a nonlinear combination of input variables an online learning algorithm which consists of structure learning and parameter learning is also presented the structure learning depends on the entropy measure to determine the number of fuzzy rules the parameter learning based on the gradient descent method can adjust the shape of the membership function and the corresponding weights of the flnn furthermore results for the universal approximator and a convergence analysis of the flnfn model are proven finally the flnfn model is applied in various simulations results of this study demonstrate the effectiveness of the proposed flnfn model

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

the cuntz toeplitz algebras have nuclear dimension one abstract we prove that unital extensions of kirchberg algebras by separable stable af algebras have nuclear dimension one the title follows

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

folner tilings for actions of amenable groups we show that every probability measure preserving action of a countable amenable group g can be tiled modulo a null set using finitely many finite subsets of g shapes with prescribed approximate invariance so that the collection of tiling centers for each shape is borel this is a dynamical version of the downarowicz huczek zhang tiling theorem for countable amenable groups and strengthens the ornstein weiss rokhlin lemma as an application we prove that for every countably infinite amenable group g the crossed product of a generic free minimal action of g on the cantor set is mathcal z stable

equivariant kirchberg phillips type absorption for amenable group actions we show an equivariant kirchberg phillips type absorption theorem for pointwise outer actions of discrete amenable groups on kirchberg algebras with respect to natural model actions on the cuntz algebras mathcal o infty and mathcal o 2 this generalizes results known for finite groups and poly mathbb z groups the model actions are shown to be determined up to strong cocycle conjugacy by natural abstract properties which are verified for some examples of actions arising from tensorial shifts we also show the following homotopy rigidity result which may be understood as a precursor to a general kirchberg phillips type classification theory if two outer actions of an amenable group on a unital kirchberg algebra are equivariantly homotopy equivalent then they are conjugate this marks the first c dynamical classification result up to cocycle conjugacy that is applicable to actions of all amenable groups

actions of certain torsion free elementary amenable groups on strongly self absorbing mathrm c c algebras in this paper we consider a bootstrap class mathfrak c of countable discrete groups which is closed under countable unions and extensions by the integers and we study actions of such groups on mathrm c algebras this class includes all torsion free abelian groups poly mathbb z groups as well as other examples using the interplay between relative rokhlin dimension and semi strongly self absorbing actions established in prior work we obtain the following two main results for any group gamma in mathfrak c and any strongly self absorbing mathrm c algebra mathcal d in fact we establish more general relative versions of these two results for actions of amenable groups that have a predetermined quotient in the class mathfrak c for the monotracial case the proof comprises an application of matui sato s equivariant property si as a key method

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

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

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

a universal multicoefficient theorem for the kasparov groups that holds in the same generality as the universal coefficient theorem of rosenberg and schochet there are advantages in some circumstances to using hom k a k b in place of kk a b these advantages derive from the fact that k a can be equipped with order and scale structures similar to those on k0 a with this additional structure the module k a becomes a powerful invariant of c algebras we show that it is a complete invariant for the class of real rank zero ad algebras the ad algebras are a certain kind of approximately subhomogeneous c algebras which may have torsion in k1 ell in addition to classifying these algebras we calculate their automorphism groups up to approximately innerautomorphisms

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

quasidiagonal traces on exact c algebras abstract recently it was proved by tikuisis white and winter that any faithful trace on a separable nuclear c algebra in the uct class is quasidiagonal building on their work we generalise the result and show that any faithful amenable trace on a separable exact c algebra in the uct class is quasidiagonal we also prove that any amenable trace on a separable exact quasidiagonal c algebra in the uct class is quasidiagonal

on the nuclear dimension of strongly purely infinite c algebras abstract we show that separable nuclear and strongly purely infinite c algebras have finite nuclear dimension in fact the value is at most three this exploits a deep structural result of kirchberg and rordam on strongly purely infinite c algebras that are homotopic to zero in an ideal system preserving way

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

quasidiagonality of nuclear c algebras we prove that faithful traces on separable and nuclear c algebras in the uct class are quasidiagonal this has a number of consequences firstly by results of many hands the classification of unital separable simple and nuclear c algebras of finite nuclear dimension which satisfy the uct is now complete secondly our result links the finite to the general version of the toms winter conjecture in the expected way and hence clarifies the relation between decomposition rank and nuclear dimension finally we confirm the rosenberg conjecture discrete amenable groups have quasidiagonal c algebras

a bivariant theory for the cuntz semigroup abstract we introduce a bivariant version of the cuntz semigroup as equivalence classes of order zero maps generalizing the ordinary cuntz semigroup the theory has many features formally analogous to kk theory including a composition product we establish basic properties like additivity stability and continuity and study categorical aspects in the setting of local c algebras we determine the bivariant cuntz semigroup for numerous examples such as when the second algebra is a kirchberg algebra and cuntz homology for compact hausdorff spaces which provides a complete invariant moreover we establish identities when tensoring with strongly self absorbing c algebras finally we show how to use the bivariant cuntz semigroup of the present work to classify unital and stably finite c algebras

central sequence c algebras and tensorial absorption of the jiang su algebra we study properties of the central sequence algebra a a 0 of a c algebra a and we present an alternative approach to a recent result of matui and sato they prove that every unital separable simple nuclear c algebra whose trace simplex is nite dimensional tensorially absorbs the jiang su algebra if and only if it has the strict comparison property we extend their result to the case where the extreme boundary of the trace simplex is closed and of nite topological dimension we are also able to relax the assumption on the c algebra of having the strict com parison property to a weaker property that we call local weak comparison namely we prove that a unital separable simple nuclear c algebra whose trace simplex has nite dimensional closed extreme boundary tensorially absorbs the jiang su algebra if and only if it has the local weak comparison property we can also eliminate the nuclearity assumption if instead we assume the si property of matui and sato and moreover that each ii1 factor representation of the c algebra is a mcdu factor

a new proof of kirchberg s 2 stable classification i present a new proof of kirchbergs q2 stable classification theorem two separable nuclear stable unital q2 stable c algebras are isomorphic if and only if their ideal lattices are order isomorphic or equivalently their primitive ideal spaces are homeomorphic many intermediate results do not depend on pure infiniteness of any sort

nuclear dimension and mathcal z stability of pure c algebras in this article i study a number of topological and algebraic dimension type properties of simple c algebras and their interplay in particular a simple c algebra is defined to be tracially m bar m pure if it has strong tracial m comparison and is tracially bar m almost divisible these notions are related to each other and to nuclear dimension

every classifiable simple c algebra has a cartan subalgebra we construct cartan subalgebras in all classifiable stably finite c algebras together with known constructions of cartan subalgebras in all uct kirchberg algebras this shows that every classifiable simple c algebra has a cartan subalgebra

an approximate universal coefficient theorem an approximate universal coefficient theorem auct for certain c algebras is established we present a proof that kirchberg phillips s classification theorem for separable nuclear purely infinite simple c algebras is valid for c algebras satisfying the auct instead of the uct it is proved that two versions of auct are in fact the same we also show that c algebras that are locally approximated by c algebras satisfying the auct satisfy the auct as an application we prove that certain simple c algebras which are locally type i are in fact isomorphic to simple ah algebras as another application we show that a sequence of residually finite dimensional c algebras which are asymptotically nuclear and which asymptotically satisfies the auct can be embedded into the same simple af algebra

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

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

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

ranks of operators in simple c algebras with stable rank one let a be a simple c algebra with stable rank one we show that every strictly positive lower semicontinuous affine function on the simplex of normalized quasitraces of a is realized as the rank of an operator in the stabilization of a assuming moreover that a has locally finite nuclear dimension we deduce that a is mathcal z stable if and only if it has strict comparison of positive elements in particular the toms winter conjecture holds for simple approximately subhomogeneous c algebras with stable rank one

the classification of rokhlin flows on mathrm c c algebras we study flows on mathrm c algebras with the rokhlin property we show that every kirchberg algebra carries a unique rokhlin flow up to cocycle conjugacy which confirms a long standing conjecture of kishimoto we moreover present a classification theory for rokhlin flows on mathrm c algebras satisfying certain technical properties which hold for many mathrm c algebras covered by the elliott program as a consequence we obtain the following further classification theorems for rokhlin flows firstly we extend the statement of kishimoto s conjecture to the non simple case up to cocycle conjugacy a rokhlin flow on a separable nuclear mathcal o infty absorbing mathrm c algebra is uniquely determined by its induced action on the prime ideal space secondly we give a complete classification of rokhlin flows on simple classifiable kk contractible mathrm c algebras two rokhlin flows on such a mathrm c algebra are cocycle conjugate if and only if their induced actions on the cone of lower semicontinuous traces are affinely conjugate

the cone of lower semicontinuous traces on a c algebra the cone of lower semicontinuous traces is studied with a view to its use as an invariant its properties include compactness hausdorffness and continuity with respect to inductive limits a suitable notion of dual cone is given the cone of lower semicontinuous 2 quasitraces on a non exact c algebra is considered as well these results are applied to the study of the cuntz semigroup it is shown that if a c algebra absorbs the jiang su algebra then the subsemigroup of its cuntz semigroup consisting of the purely non compact elements is isomorphic to the dual cone of the cone of lower semicontinuous 2 quasitraces this yields a computation of the cuntz semigroup for the following two classes of c algebras c algebras that absorb the jiang su algebra and have no non zero simple subquotients and simple c algebras that absorb the jiang su algebra 1 introduction the most standard invariants in the classification of nu clear c algebras are their k groups and their traces the traces at least for simple c algebras are assumed to be bounded in the unital case and lower semicontinuous and densely finite in the non unital case if one has in mind the classification of non simple c algebras it is clear that these two kinds of traces will not suffice and a broader class should be considered in this paper we study the properties of the cone of all lower semicontinuous traces on a c algebra with the purpose of applying our results to questions in the classification of non simple c algebras one sign that this is natural is that this cone in particular determines the lattice of closed two sided ideals an important invariant in its own right see 3 4 we shall also consider lower semicontinuous 2 quasitraces since these appear naturally as functionals on the cuntz semigroup of the algebra if the