minimal dynamical systems on the product of the cantor set and the circle we prove that a crossed product algebra arising from a minimal dynamical system on the product of the cantor set and the circle has real rank zero if and only if the set of invariant measures of the system come from the associated cantor minimal system in the case that cocycles take values in the rotation group it is also shown that this condition implies tracial rank zero and in particular the crossed product algebra is isomorphic to a unital simple at algebra of real rank zero under the same assumption we show that two systems are approximately k conjugate if and only if there exists a sequence of isomorphisms between two associated crossed products which approximately maps open image in new window

on tracial approximation let c be a class of unital c algebras the class tac of c algebras which can be tracially approximated in the egorov like sense first considered by lin by the c algebras in c is studied lin considered the case that c consists of finite dimensional c algebras or the tensor products of such with c 0 1 in particular the question is considered whether for any simple separable a tac there is a c algebra b which is a simple inductive limit of certain basic homogeneous c algebras together with c algebras in c such that the elliott invariant of a is isomorphic to the elliott invariant of b an interesting case of this question is answered in the final part of the paper the question is also considered which properties of c algebras are inherited by tracial approximation results of this kind are obtained which are used in the proof of the main theorem of the paper and also in the proof of the classification theorem of the second author given in z niu a classification of tracially approximately splitting tree algebra in preparation and z niu a classification of certain tracially approximately subhomogeneous c algebras phd thesis university of toronto 2005 which also uses the main result of the present paper

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

fundamental group of simple c algebras with unique trace abstract we introduce the fundamental group f a of a unital simple c algebra a with a unique normalized trace we compute fundamental groups f a of several nuclear or non nuclear c algebras a k theoretical obstruction enables us to compute the fundamental group easily our study is essentially based on the computation of picard groups by kodaka

towards a theory of classification abstract the well known difficulties arising in a classification which is not set theoretically trivial involving what is sometimes called a non smooth quotient have been overcome in a striking way in the theory of operator algebras by the use of what might be called a classification functor the very existence of which is already a surprise here the notion of such a functor is developed abstractly and a number of examples are considered including those which have arisen for various classes of operator algebras

the number of limit cycles of certain polynomial differential equations two dimensional differential systems are considered where p and q are polynomials the question of interest is the maximum possible numberof limit cycles of such systems in terms of the degree of p and q an algorithm is described for determining a so called focal basis this can be implemented on a computer estimates can then be obtained for the number of small amplitude limit cycles the technique is applied to certain cubic systems a class of examples with exactly five small amplitude limit cycles is constructed quadratic systems are also considered

the nuclear dimension of graph c algebras abstract consider a graph c algebra c e with a purely infinite ideal i possibly all of c e such that i has only finitely many ideals and c e i is approximately finite dimensional we prove that the nuclear dimension of c e is 1 if i has infinitely many ideals then the nuclear dimension of c e is either 1 or 2

classifying c algebras via ordered mod p k theory we introduce an order structure on k 0 9 k0 z p this group may also be thought of as ko 7z z p we exhibit new examples of real rank zero c algebras that are inductive limits of finite dimensional and dimension drop algebras have the same ordered graded k theory with order unit and yet are not isomorphic in fact they are not even stably shape equivalent the order structure on k0 z 3 z p naturally distinguishes these algebras the same invariant is used to give an isomorphism theorem for such realrank zero inductive limits as a corollary we obtain an isomorphism theorem for all real rank zero approximately homogeneous c algebras that arise from systems of bounded dimension growth and torsion free k0 group at the 1980 kingston conference effros posed the problem of finding suitable invariants for use in studying c algebras that are limits of sequences of homogeneous c algebras these are now called almost homogeneous ah c algebras the classification of ah algebras is a rapidly developing field and we will not attempt to summarize all this activity instead we will focus on the growth of the invariants used specifically we consider an ah algebra a that is the direct limit o f a system of the form

a class of simple c algebras with stable rank one abstract we study the limits of inductive sequences a i i where each a i is a direct sum of full matrix algebras over compact metric spaces and each partial map of i is diagonal we give a new characterisation of simplicity for such algebras and apply it to prove that the said algebras have stable rank one whenever they are simple and unital significantly our results do not require any dimension growth assumption

extensions of c x by simple c algebras of real rank zero let ext c x a be the set of unitarily equivalence classes of essential c algebra exten sions of the following form 0 a e c x 0 where a is a nonunital separable simple c algebra of real rank zero stable rank one with unique normalized trace and x is a finite cw complex we show that there is a bijection ext c x a kk c x m a a where m a is the multiplier algebra of a in particular we determine when an extension is actually splitting we also in a more general setting give a condition when an essential extension is quasidiagonal 0 introduction we study essential extensions of the following form 0 a f c x 0 where a is a j unital nonunital simple c algebra of real rank zero and stable rank one and x is finite cw complex the brown douglas fillmore bdf theory bdf1 and bdf2 gives a classification of these extensions up to unitary equivalence in the case that a k the c algebra of compact operators on an infinite dimensional separable hubert space since then the theory of c algebra extensions developed rapidly it is very difficult to give a complete list we refer the reader to bl for references while using of theory ext c x a have been computed for the most interesting cases however ext c x a does not provide enough information in general to classify extensions up to unitary equivalence in fact as shown in ln6 and lnl2 there are usually many different unitary equivalence classes of trivial extensions and an extension r with r 0 in ext c x a may not be trivial we will only consider simple c algebras that share some properties with c we consider only those with real rank zero stable rank one and unique quasi trace up to scalar multiples there is an additional property that c has namely its corona algebra m k k the calkin algebra is simple we also as sume that m a a is simple it has been studied when a nonunital but cr unital simple c algebra a has the property that m a a is simple see lnl rrl and zh2 etc it is shown in lnl that a simple c algebra a has the property that m a a is simple if and only if a has continuous scale

theory of strongly irreducible operators and its applications in the matrix theory of finite dimensional space the famous jordan canonical form theorem sufficiently reveals the internal structure of matrices in the survey we will give similar jordan canonical form theorems of operator theory in a complex separable infinite dimensional hilbert space by the way we also show their applications in other mathematical branches

recursive subhomogeneous algebras we introduce and characterize a particularly tractable class of unital type 1 c algebras with bounded dimension of irreducible representations algebras in this class are called recursive subhomogeneous algebras and they have an inductive description through iterated pullbacks which allows one to carry over from algebras of the form many of the constructions relevant in the study of the stable rank and k theory of simple direct limits of homogeneous c algebras our characterization implies in particular that if is a separable c algebra whose irreducible representations all have dimension at most and if for each the space of dimensional irreducible representations has finite covering dimension then is a recursive subhomogeneous algebra we demonstrate the good properties of this class by proving subprojection and cancellation theorems in it consequences for simple direct limits of recursive subhomogeneous algebras with applications to the transformation group c algebras of minimal homeomorphisms will be given in separate papers

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

inductive limits of interval algebras the tracial state space introduction this is the first in a series of papers where we investigate the structure of the c algebras obtained as inductive limits of interval algebras c 0 1 0m the purpose here is to show that these c algebras comprise a large stock of simple c algebras and that some of them allow endomorphisms with chaotic actions on the tracial state space to be more specific we show that to any dense subgroup g of q containing z and any metrizable choquet simplex s there is a sequence

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

convex subshifts separated bratteli diagrams and ideal structure of tame separated graph algebras abstract we introduce a new class of partial actions of free groups on totally disconnected compact hausdorff spaces which we call convex subshifts these serve as an abstract framework for the partial actions associated with finite separated graphs in much the same way as classical subshifts generalize the edge shift of a finite graph we define the notion of a finite type convex subshift and show that any such subshift is kakutani equivalent to the partial action associated with a finite bipartite separated graph we then study the ideal structure of both the full and the reduced tame graph c algebras o e c and o r e c of a separated graph e c and of the abelianized leavitt path algebra l k ab e c as well these algebras are the reduced crossed products with respect to the above mentioned partial actions and we prove that there is a lattice isomorphism between the lattice of induced ideals and the lattice of hereditary d saturated subsets of a certain infinite separated graph f d built from e c called the separated bratteli diagram of e c we finally use these tools to study simplicity and primeness of the tame separated graph algebras

classification of simple c algebras with unique traces we classify certain almost multiplicative morphisms up to approximate unitary equivalence and use this result to prove the following let a and b be two unital separable simple c algebras of real rank zero stable rank one with weakly unperforated k 0 groups and with unique normalized quasi traces suppose that both a and b are locally ah and k a k a 1 a k b k b 1 b then a is isomorphic to b

finite group actions on c algebras with the rohlin property i basic properties of finite group actions with the rohlin property on unital c algebras are investigated a characterization of finite group actions with the rohlin property on the cuntz algebra 02 is given in terms of central sequences which may be considered as an equivariant version of e kirchberg and n c phillips s characterization of 02 a large class of symmetries on 02 are classified in terms of the fixed point algebras for conjugacy and the crossed products for cocycle conjugacy model actions of symmetries of 02 are constructed for given k theoretical 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

asymptotically unitary equivalence and asymptotically inner automorphisms let c be a unital ah algebra and let a be a unital separable simple c algebra with tracial rank zero suppose that phi 1 phi 2 colon c to a are two unital monomorphisms we show that there is a continuous path of unitaries u t colon t in 0 infty of a such that lim t to infty u t phi 1 a u t phi 2 a quad hbox rm for all quad a in c if and only if phi 1 phi 2 in kk c a tau circ phi 1 tau circ phi 2 for all tau in t a and the rotation map tilde eta phi 1 phi 2 associated with phi 1 and phi 2 is zero in particular an automorphism alpha on a unital separable simple c algebra a in cal n with tracial rank zero is asymptotically inner if and only if alpha rm id a quad rm in quad kk a a and the rotation map tilde eta phi 1 phi 2 is zero let a be a unital ah algebra not necessarily simple and let alpha in aut a be an automorphism as an application we show that the associated crossed product a rtimes alpha bbb z can be embedded into a unital simple af algebra if and only if a admits a strictly positive alpha invariant tracial state

approximate homogeneity is not a local property it is shown that the ah algebras satisfy a certain splitting property at the level of k theory with torsion coefficients the splitting property is used to prove the following i there are locally homogeneous c algebras which are not ah algebras ii the class of ah algebras is not closed under countable inductive limits iii there are real rank zero split quasidiagonal extensions of ah algebras which are not ah algebras

cut down method in the inductive limit decomposition of non commutative tori iii a complete answer in 3 dimension we use the cut down method and a multi dimensional continued fraction approximation to prove that any simple 3 torus is an inductive limit of direct sums of four circle algebras consequently simple 3 tori are classified by the orderedk0 group with distinguished order unit

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

mathbb z actions on ah algebras and mathbb z 2 actions on af algebras we consider mathbb z actions single automorphisms on a unital simple ah algebra with real rank zero and slow dimension growth and show that the uniform outerness implies the rohlin property under some technical assumptions moreover two mathbb z actions with the rohlin property on such a c algebra are shown to be cocycle conjugate if they are asymptotically unitarily equivalent we also prove that locally approximately inner and uniformly outer mathbb z 2 actions on a unital simple af algebra with a unique trace have the rohlin property and classify them up to cocycle conjugacy employing the orderext group as classification invariants

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

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

a second order accurate immersed boundary lattice boltzmann method for particle laden flows abstract a new immersed boundary lattice boltzmann method ib lbm is presented for fully resolved simulations of incompressible viscous flows laden with rigid particles the immersed boundary method ibm recently developed by breugem 2012 19 is adopted in the present method development including the retraction technique the multi direct forcing method and the direct account of the inertia of the fluid contained within the particles the present ib lbm is however formulated with further improvement with the implementation of the high order runge kutta schemes in the coupled fluid particle interaction the major challenge to implement high order runge kutta schemes in the lbm is that the flow information such as density and velocity cannot be directly obtained at a fractional time step from the lbm since the lbm only provides the flow information at an integer time step this challenge can be however overcome as given in the present ib lbm by extrapolating the flow field around particles from the known flow field at the previous integer time step the newly calculated fluid particle interactions from the previous fractional time steps of the current integer time step are also accounted for in the extrapolation the ib lbm with high order runge kutta schemes developed in this study is validated by several benchmark applications it is demonstrated for the first time that the ib lbm has the capacity to resolve the translational and rotational motion of particles with the second order accuracy the optimal retraction distances for spheres and tubes that help the method achieve the second order accuracy are found to be around 0 30 and 0 47 times of the lattice spacing respectively simulations of the stokes flow through a simple cubic lattice of rotational spheres indicate that the lift force produced by the magnus effect can be very significant in view of the magnitude of the drag force when the practical rotating speed of the spheres is encountered this finding may lead to more comprehensive studies of the effect of the particle rotation on fluid solid drag laws it is also demonstrated that when the third order or the fourth order runge kutta scheme is used the numerical stability of the present ib lbm is better than that of all methods in the literature including the previous ib lbms and also the methods with the combination of the ibm and the traditional incompressible navier stokes solver

the classification of certain non simple c algebras of tracial rank zero abstract it is proved that the z 2 graded ordered k theory group with order unit k a k a 1 a is a complete invariant for the class of unital c algebras which are inductive limits of finite direct sums of unital amenable separable simple c algebras with tracial rank zero which satisfy the uct

the automorphism group of a simple tracially ai algebra the structure of the automorphism group of a simple tai algebra is studied in particular we show that frac overline rm inn a overline rm inn 0 a is isomorphic as a topological group to an inverse limit of discrete abelian groups for a unital simple ah algebra with bounded dimension growth consequently frac overline rm inn a overline rm inn 0 a is totally disconnected

crossed products by minimal homeomorphisms let x be an infinite compact metric space with finite covering dimension and let h x x be a minimal homeomorphism we show that the associated crossed product c algebra a 1 4 c z x h has tracial rank zero whenever the image of k0 a in a t a is dense as a consequence we show that these crossed product c algebras are in fact simple ah algebras with real rank zero when x is connected and h is further assumed to be uniquely ergodic then the above happens if and only if the rotation number associated to h has irrational values by applying the classification result for nuclear simple c algebras with tracial rank zero we show that two such dynamical systems have isomorphic crossed products if and only if they have isomorphic scaled ordered k theory

relative positions of matroid algebras abstract a classification is given for regular positions d d d of jones index 4 where equation omitted is an even matroid algebra and where the individual summands have index 2 a similar classification is obtained for positions of direct sums of 2 symmetric algebras and in the odd case for the positions of sums of 2 symmetric c algebras in matroid c algebras the approach relies on an analysis of intermediate non self adjoint operator algebras and the classifications are given in terms of k0 invariants partial isometry homology and scales in the composite invariant k0 h1

isomorphism and morita equivalence classes for crossed products of irrational rotation algebras by cyclic subgroups of sl2 z abstract let be irrational numbers and a b be matrices in s l 2 z of infinite order we compute the k theory of the crossed product a a z and show that a a z and a b z are isomorphic if and only if mod z and i a 1 is matrix equivalent to i b 1 combining this result and an explicit construction of equivariant bimodules we show that a a z and a b z are morita equivalent if and only if and are in the same g l 2 z orbit and i a 1 is matrix equivalent to i b 1 finally we determine the morita equivalence class of a f for any finite subgroup f of s l 2 z

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

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

when almost multiplicative morphisms are close to homomorphisms absrract it is shown that approximately multiplicative contractive positive morphisms from c x with dim x 2 into a simple c algebra a of real rank zero and of stable rank one are close to homomorphisms provided that certain k theoretical obstacles vanish as a corollary we show that a homomorphism h c x a is approximated by homomorphisms with finite dimensional range if h gives no k theoretical obstacle

on inductive limits of matrix algebras over the two torus it will be shown in this paper that certain real rank zero c algebras which are inductive limits of c algebras of the form i m k i c inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i can be expressed as inductive limits of c algebras of the form i m k i c s 1 in particular if both a and b are of real rank zero and are inductive limits of c algebras of the form i m k i c s 1 then also a b is an inductive limit of c algebras of the form i m k i c s 1 hence a b can be classified by its k theory this is a key step in the general classification theory of inductive limit c algebras

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