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

ranks of operators in simple c algebras let a be a unital simple separable c algebra with strict comparison of positive elements we prove that the cuntz semigroup of a is recovered functorially from the murray von neumann semigroup and the tracial state space t a whenever the extreme boundary of t a is compact and of finite covering dimension combined with a result of winter we obtain z a a whenever a moreover has locally finite decomposition rank as a corollary we confirm elliott s classification conjecture under reasonably general hypotheses which notably do not require any inductive limit structure these results all stem from our investigation of a basic question what are the possible ranks of operators in a unital simple c algebra

inductive limits of subhomogeneous c algebras with hausdorff spectrum we consider unital simple inductive limits of generalized dimension drop c algebras they are so called ash algebras and include all unital simple ah algebras and all dimension drop c algebras suppose that a is one of these c algebras we show that a q has tracial rank no more than one where q is the rational uhf algebra as a consequence we obtain the following classification result let a and b be two unital simple inductive limits of generalized dimension drop algebras with no dimension growth then a b if and only if they have the same elliott invariant

the range of approximate unitary equivalence classes of homomorphisms from ah algebras let c be a unital ah algebra and a be a unital simple c algebras with tracial rank zero it has been shown that two unital monomorphisms phi psi c to a are approximately unitarily equivalent if and only if phi psi quad rm in quad kl c a quad rm and quad tau circ phi tau circ psi quad rm for all tau in t a where t a is the tracial state space of a in this paper we prove the following given kappa in kl c a with kappa k 0 c setminus 0 subset k 0 a setminus 0 and with 1c 1a and a continuous affine map lambda t a to t mathfrak f c which is compatible with where t mathfrak f c is the convex set of all faithful tracial states there exists a unital monomorphism phi c to a such that phi kappa quad rm and quad tau circ phi c lambda tau c for all c in c s a and tau in t a denote by rm mon au e c a the set of approximate unitary equivalence classes of unital monomorphisms we provide a bijective map lambda rm mon au e c a to klt c a where klt c a is the set of compatible pairs of elements in kl c a and continuous affine maps from t a to t mathfrak f c moreover we found that there are compact metric spaces x unital simple af algebras a and kappa in kl c x a with kappa k 0 c x setminus 0 subset k 0 a setminus 0 for which there is no homomorphism h c x a so that h

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

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

a classification of simple limits of splitting interval algebras abstract let a be a unital simple limit of finite direct sums of sub homogeneous interval algebras of a certain type cf definition 1 1 it is proved that a can be classified by the scaled ordered group k 0 a the simplex t a and the canonical pairing between them it is also shown that k 0 a might fail to have the riesz decomposition property

the ext class of an approximately inner automorphism let a be a simple unital at algebra of real rank zero it is shown below that the range of the natural map from the approximately inner automorphism group to kk a a coincides with the kernel of the map kk a a l 1 0 hom ki a ki a 1 introduction and preliminaries 1 1 an automorphism of a unital c algebra a is said to be an approximately inner automorphism if there is a sequence of unitaries un 2 a such that a limadun a for all a 2 a it follows that the induced map on k a is the identity map there is however an invariant of k theoretical interest which can occur nontrivial extensions may arise in the six term periodic sequence for the k theory of the mapping torus we show below that every extension does arise if a is a simple unital at algebra of real rank zero as an immediate corollary we obtain a stronger form of elliott s classification theorem for such algebras an invertible kk element that preserves positivity and the class of the unit lifts to an isomorphism 1 2 recall that a unital c algebra a is said to be a unital at algebra if it is expressible as the inductive limit of finite direct sums of algebras of the form mn c t with unital embeddings let a be a unital at algebra and let be an approximately inner automorphism of a the mapping torus of is the c algebra

finitely generated nilpotent group c algebras have finite nuclear dimension we show that group c algebras of finitely generated nilpotent groups have finite nuclear dimension it then follows from a string of deep results that the c algebra a generated by an irreducible representation of such a group has decomposition rank at most 3 if in addition a satisfies the universal coefficient theorem another string of deep results shows it is classifiable by its ordered ktheory and is approximately subhomogeneous we observe that all c algebras generated by faithful irreducible representations of finitely generated torsion free nilpotent groups satisfy the universal coefficient theorem

asymptotic unitary equivalence and classification of simple amenable c algebras let c and a be two unital separable amenable simple c algebras with tracial rank at most one suppose that c satisfies the universal coefficient theorem and suppose that 1 2 c a are two unital monomorphisms we show that there is a continuous path of unitaries u t t 0 of a such that lim t to infty u t varphi 1 c u t varphi 2 c quad mbox for all c in c if and only if 1 2 in kk c a varphi 1 ddag varphi 2 ddag varphi 1 t varphi 2 t and a rotation related map overline r varphi 1 varphi 2 associated with 1 and 2 is zero applying this result together with a result of w winter we give a classification theorem for a class mathcal a of unital separable simple amenable c algebras which is strictly larger than the class of separable c algebras with tracial rank zero or one tensor products of two c algebras in mathcal a are again in mathcal a moreover this class is closed under inductive limits and contains all unital simple ash algebras for which the state space of k 0 is the same as the tracial state space and also some unital simple ash algebras whose k 0 group is and whose tracial state spaces are any metrizable choquet simplex one consequence of the main result is that all unital simple ah algebras which are mathcal z stable are isomorphic to ones with no dimension growth

on 2k variable symmetric boolean functions with maximum algebraic immunity k given a positive even integer n it is found that the weight distribution of any n variable symmetric boolean function with maximum algebraic immunity ai n 2 is determined by the binary expansion of n based on the foregoing all n variable symmetric boolean functions with maximum ai are constructed the amount is 2 wt n 1 2 log2n

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

classification of c algebras generated by representations of the unitriangular group ut 4 z abstract it was recently shown that each c algebra generated by a faithful irreducible representation of a finitely generated torsion free nilpotent group is classified by its ordered k theory for the three step nilpotent group u t 4 z we calculate the ordered k theory of each c algebra generated by a faithful irreducible representation of u t 4 z and see that they are all simple a t algebras we also point out that there are many simple non a t algebras generated by irreducible representations of nilpotent groups

the classification of simple separable unital z stable locally ash algebras let a be a unital simple separable locally approximately subhomogeneous c algebra locally ash algebra it is shown that a qa q can be tracially approximated by unital elliott thomsen algebras with trivial k1k1 group where q is the universal uhf algebra in particular it follows that a is classifiable by the elliott invariant if a is jiang su stable

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

k theoretic rigidity and slow dimension growth let a be an approximately subhomogeneous ash c algebra with slow dimension growth we prove that if a is unital and simple then the cuntz semigroup of a agrees with that of its tensor product with the jiang su algebra mathcal z in tandem with a result of w winter this yields the equivalence of mathcal z stability and slow dimension growth for unital simple ash algebras this equivalence has several consequences including the following classification theorem unital ash algebras which are simple have slow dimension growth and in which projections separate traces are determined up to isomorphism by their graded ordered k theory and none of the latter three conditions can be relaxed in general

exponentials in simple z stable c algebras abstract let a be a unital separable simple z stable c algebra which has rational tracial rank at most one and let u u 0 a where u 0 a is the connected component of the unitary group of a containing the identity we show that for any 0 there exists a self adjoint element h a such that 0 1 u exp i h but there is no control of h in general let cu a be the closure of the commutator subgroup of the unitary group of a and let u cu a we prove that there exists a self adjoint element h a such that 0 2 u exp i h and h 2 examples are given that the bound 2 for h is optimal in general for the jiang su algebra z we show that if u u 0 z and 0 there exists a real number t and a self adjoint element h z with h 2 such that e i t u exp i h

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

simple nuclear c algebras of tracial topological rank one we give a classification theorem for unital separable nuclear c algebras with tracial rank no more than one let a and b be two unital separable simple nuclear c algebras with tr a tr b 1 which satisfy the universal coefficient theorem we show that a b if and only if there is an order and unit preserving isomorphism 0 1 2 k0 a k0 a 1a k1 a t a k0 b k0 b 1b k1 b t b where 2 1 x 0 x for each x k0 a and t b

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

classification of extensions of classifiable c algebras abstract for a certain class of extensions e 0 b e a 0 of c algebras in which b and a belong to classifiable classes of c algebras we show that the functor which sends e to its associated six term exact sequence in k theory and the positive cones of k 0 b and k 0 a is a classification functor we give two independent applications addressing the classification of a class of c algebras arising from substitutional shift spaces on one hand and of graph algebras on the other the former application leads to the answer of a question of carlsen and the first named author concerning the completeness of stabilized matsumoto algebras as an invariant of flow equivalence the latter leads to the first classification result for nonsimple graph c algebras

minimal dynamics and the classification of c algebras let x be an infinite compact metrizable space of finite covering dimension and x x a minimal homeomorphism we prove that the crossed product c x absorbs the jiang su algebra tensorially and has finite nuclear dimension as a consequence these algebras are determined up to isomorphism by their graded ordered k theory under the necessary condition that their projections separate traces this result applies in particular to those crossed products arising from uniquely ergodic homeomorphisms

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

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

edge representation with fuzzy sets in blurred images this paper proposes a representation of edges in blurred images using the concept of fuzzy sets when the images are degraded by various asymmetric and local blurring factors the proposed representation is expressed by fuzzy membership functions and it can serve as a relative index of blur the membership function is derived from the distribution of intensity gradients and the symmetricity of gradient magnitudes and the function is calculated directly from the blurred image without identifying the point spread function or restoring image in this way the fuzzy edge representation describes edges with their degradation states by fuzzy memberships instead of the binary description of edges the index of fuzziness reflects the average amount of ambiguity presented in a fuzzy set and the moments denote an object in the image quantitatively these measures are adopted to illustrate the effectiveness of the representation in locally blurred images

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

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