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

on the domain of hyperbolicity of the cumulant equations in this article we consider the influence of non equilibirum values of classical variables on the eigenvalues of the advection part of the cumulant equations real and finite eigenvalues are a neccessary condition for the cumulant equations to be hyperbolic which can be used to obtain estimates on admissible deviations from equilibrium for a model of particular order still to be valid we find that this condition puts no constraints on velocity and shear stress values but specific energy must be positive normal stress must be bounded by specific energy and heat flux not be too large

symmetries of the car algebra a automorphism of period two of the uhf c algebra of type 2 is constructed for which the fixed point algebra is not approximately finite dimensional the construction also yields a diagonal maximal commutative c subalgebra whose spectrum is not totally disconnected variations and consequences of the construction are discussed

the boltzmann equation for weakly inhomogeneous data we solve the initial value problem associated to the nonlinear boltzmann equation in the case in which the initial distribution has sufficiently small spatial gradients

mean dimension and ah algebras with diagonal maps mean dimension for ah algebras with diagonal maps is introduced it is shown that if a simple unital ah algebra with diagonal maps has mean dimension zero then it has strict comparison on positive elements in particular the strict order on projections is determined by traces moreover a lower bound of the mean dimension is given in term of toms comparison radius using classification results if a simple unital ah algebra with diagonal maps has mean dimension zero it must be an ah algebra without dimension growth two classes of ah algebras with diagonal maps are shown to have mean dimension zero the class of ah algebras with at most countably many extremal traces and the class of ah algebras with numbers of extreme traces which induce same state on the k0 group being uniformly bounded in particular this class includes ah algebras with real rank zero

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

complete orthogonality preservers between c algebras abstract we introduce n orthogonality and completely orthogonality preserving operators between c algebras our main theorem states that every completely orthogonality preserving bounded linear mapping between c algebras is a weighted tro homomorphism we also give several characterisations of tro homomorphisms among triple homomorphisms

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

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

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

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

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

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

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

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

dimension growth for c algebras we introduce the growth rank of a c algebra a valued invariant whose minimal instance is equivalent to the condition that an algebra absorbs the jiang su algebra tensorially and prove that its range is exhausted by simple nuclear c algebras as consequences we obtain a well developed theory of dimension growth for approximately homogeneous ah c algebras establish the existence of simple nuclear and non stable c algebras which are not tensorially prime and show the assumption of stability to be particularly natural when seeking classification results for nuclear c algebras via k theory the properties of the growth rank lead us to propose a universal property which can be considered inside any class of unital and nuclear c algebras we prove that satisfies this universal property inside a large class of locally subhomogeneous algebras representing the first uniqueness theorem for which does not depend on the classification theory of nuclear 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

flat dimension growth for c algebras abstract simple and nuclear c algebras which fail to absorb the jiang su algebra tensorially have settled many open questions in the theory of nuclear c algebras but have been little studied in their own right this is due partly to a dearth of invariants sensitive to differences between such algebras we present two new real valued invariants to fill this void the dimension rank ratio for unital ah algebras and the radius of comparison for unital and stably finite algebras we establish their basic properties show that they have natural connections to ordered k theory and prove that the range of the dimension rank ratio is exhausted by simple algebras this last result shows the class of simple nuclear and non z stable c algebras to be uncountable in passing we establish a theory of moderate dimension growth for ah algebras the existence of which was first supposed by blackadar the minimal instances of both invariants are shown to coincide with the condition of being tracially af among simple unital ah algebras of real rank zero and stable rank one whence they may be thought of as generalised measures of dimension growth we argue that the radius of comparison may be thought of as an abstract version of the dimension rank ratio

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

applications of model theory to c dynamics we initiate the study of compact group actions on c algebras from the perspective of model theory and present several applications to c dynamics firstly we prove that the continuous part of the central sequence algebra of a strongly self absorbing action is indistinguishable from the continuous part of the sequence algebra and in fact equivariantly isomorphic under the continuum hypothesis as another application we present a unified approach to several dimensional inequalities in c algebras which is done through the notion of order zero dimension for an equivariant homomorphism finiteness of the order zero dimension implies that the dimension of the target algebra can be bounded by the dimension of the domain the dimension can be among others decomposition rank nuclear dimension or rokhlin dimension as a consequence we obtain new inequalities for these quantities as a third application we obtain the following result if a c algebra a absorbs a strongly self absorbing c algebra d and is an action of a compact group g on a with finite rokhlin dimension with commuting towers then absorbs any strongly self absorbing action of g on d this has a number of interesting consequences already in the case of the trivial action on d for example we deduce that d stability passes from a to the crossed product additionally in many cases of interest our result restricts the possible values of the rokhlin dimension to and showing a striking parallel to the behavior of the nuclear dimension for simple c algebras we also show that an action of a finite group with finite rokhlin dimension with commuting towers automatically has the rokhlin property if the algebra is uhf absorbing

the jiang su algebra revisited we give a number of new characterizations of the jiang su algebra z both intrinsic and extrinsic in terms of c algebraic dynamical topological and k theoretic conditions along the way we study divisibility properties of c algebras we give a precise characterization of those unital c algebras of stable rank one that admit a unital embedding of the dimensiondrop c algebra zn n 1 and we prove a cancellation theorem for the cuntz semigroup of c algebras of stable rank one

moment closure hierarchies for kinetic theories this paper presents a systematicnonperturbative derivation of a hierarchy of closed systems of moment equations corresponding to any classical kinetic theory the first member of the hierarchy is the euler system which is based on maxwellian velocity distributions while the second member is based on nonisotropic gaussian velocity distributions the closure proceeds in two steps the first ensures that every member of the hierarchy is hyperbolic has an entropy and formally recovers the euler limit the second involves modifying the collisional terms so that members of the hierarchy beyound the second also recover the correct navier stokes behavior this is achieved through the introduction of a generalization of the bgk collision operator the simplest such system in three spatial dimensions is a 14 moment closure which also recovers the behavior of the grad 13 moment system when the velocity distributions lie near local maxwellians the closure procedure can be applied to a general class of kinetic theories

ordered c ast modules in this first part of a study of ordered operator spaces we develop the basic theory of ordered c bimodules a crucial role is played by open tripotents a jb triple variant of akemann s notion of open projection

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

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

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