CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
Canadian Mathematical Society
  location:  Publicationsjournals
Publications        
Search results

Search: MSC category 46L35 ( Classifications of $C^$-algebras *$-algebras * )

  Expand all        Collapse all Results 1 - 15 of 15

1. CJM Online first

Lin, Huaxin
Minimal Dynamical Systems on Connected Odd Dimensional Spaces
Let $\beta\colon S^{2n+1}\to S^{2n+1}$ be a minimal homeomorphism ($n\ge 1$). We show that the crossed product $C(S^{2n+1})\rtimes_\beta \mathbb{Z}$ has rational tracial rank at most one. Let $\Omega$ be a connected compact metric space with finite covering dimension and with $H^1(\Omega, \mathbb{Z})=\{0\}.$ Suppose that $K_i(C(\Omega))=\mathbb{Z}\oplus G_i,$ where $G_i$ is a finite abelian group, $i=0,1.$ Let $\beta\colon \Omega\to\Omega$ be a minimal homeomorphism. We also show that $A=C(\Omega)\rtimes_\beta\mathbb{Z}$ has rational tracial rank at most one and is classifiable. In particular, this applies to the minimal dynamical systems on odd dimensional real projective spaces. This is done by studying minimal homeomorphisms on $X\times \Omega,$ where $X$ is the Cantor set.

Keywords:minimal dynamical systems
Categories:46L35, 46L05

2. CJM 2013 (vol 65 pp. 1287)

Reihani, Kamran
$K$-theory of Furstenberg Transformation Group $C^*$-algebras
The paper studies the $K$-theoretic invariants of the crossed product $C^{*}$-algebras associated with an important family of homeomorphisms of the tori $\mathbb{T}^{n}$ called Furstenberg transformations. Using the Pimsner-Voiculescu theorem, we prove that given $n$, the $K$-groups of those crossed products, whose corresponding $n\times n$ integer matrices are unipotent of maximal degree, always have the same rank $a_{n}$. We show using the theory developed here that a claim made in the literature about the torsion subgroups of these $K$-groups is false. Using the representation theory of the simple Lie algebra $\frak{sl}(2,\mathbb{C})$, we show that, remarkably, $a_{n}$ has a combinatorial significance. For example, every $a_{2n+1}$ is just the number of ways that $0$ can be represented as a sum of integers between $-n$ and $n$ (with no repetitions). By adapting an argument of van Lint (in which he answered a question of Erdős), a simple, explicit formula for the asymptotic behavior of the sequence $\{a_{n}\}$ is given. Finally, we describe the order structure of the $K_{0}$-groups of an important class of Furstenberg crossed products, obtaining their complete Elliott invariant using classification results of H. Lin and N. C. Phillips.

Keywords:$K$-theory, transformation group $C^*$-algebra, Furstenberg transformation, Anzai transformation, minimal homeomorphism, positive cone, minimal homeomorphism
Categories:19K14, 19K99, 46L35, 46L80, , 05A15, 05A16, 05A17, 15A36, 17B10, 17B20, 37B05, 54H20

3. CJM 2013 (vol 66 pp. 596)

Eilers, Søren; Restorff, Gunnar; Ruiz, Efren
The Ordered $K$-theory of a Full Extension
Let $\mathfrak{A}$ be a $C^{*}$-algebra with real rank zero which has the stable weak cancellation property. Let $\mathfrak{I}$ be an ideal of $\mathfrak{A}$ such that $\mathfrak{I}$ is stable and satisfies the corona factorization property. We prove that $ 0 \to \mathfrak{I} \to \mathfrak{A} \to \mathfrak{A} / \mathfrak{I} \to 0 $ is a full extension if and only if the extension is stenotic and $K$-lexicographic. {As an immediate application, we extend the classification result for graph $C^*$-algebras obtained by Tomforde and the first named author to the general non-unital case. In combination with recent results by Katsura, Tomforde, West and the first author, our result may also be used to give a purely $K$-theoretical description of when an essential extension of two simple and stable graph $C^*$-algebras is again a graph $C^*$-algebra.}

Keywords:classification, extensions, graph algebras
Categories:46L80, 46L35, 46L05

4. CJM 2011 (vol 64 pp. 544)

Li, Zhiqiang
On the Simple Inductive Limits of Splitting Interval Algebras with Dimension Drops
A K-theoretic classification is given of the simple inductive limits of finite direct sums of the type I $C^*$-algebras known as splitting interval algebras with dimension drops. (These are the subhomogeneous $C^*$-algebras, each having spectrum a finite union of points and an open interval, and torsion $\textrm{K}_1$-group.)

Categories:46L05, 46L35

5. CJM 2011 (vol 64 pp. 573)

Nawata, Norio
Fundamental Group of Simple $C^*$-algebras with Unique Trace III
We introduce the fundamental group ${\mathcal F}(A)$ of a simple $\sigma$-unital $C^*$-algebra $A$ with unique (up to scalar multiple) densely defined lower semicontinuous trace. This is a generalization of ``Fundamental Group of Simple $C^*$-algebras with Unique Trace I and II'' by Nawata and Watatani. Our definition in this paper makes sense for stably projectionless $C^*$-algebras. We show that there exist separable stably projectionless $C^*$-algebras such that their fundamental groups are equal to $\mathbb{R}_+^\times$ by using the classification theorem of Razak and Tsang. This is a contrast to the unital case in Nawata and Watatani. This study is motivated by the work of Kishimoto and Kumjian.

Keywords:fundamental group, Picard group, Hilbert module, countable basis, stably projectionless algebra, dimension function
Categories:46L05, 46L08, 46L35

6. CJM 2011 (vol 63 pp. 381)

Ji, Kui ; Jiang, Chunlan
A Complete Classification of AI Algebras with the Ideal Property
Let $A$ be an AI algebra; that is, $A$ is the $\mbox{C}^{*}$-algebra inductive limit of a sequence $$ A_{1}\stackrel{\phi_{1,2}}{\longrightarrow}A_{2}\stackrel{\phi_{2,3}}{\longrightarrow}A_{3} \longrightarrow\cdots\longrightarrow A_{n}\longrightarrow\cdots, $$ where $A_{n}=\bigoplus_{i=1}^{k_n}M_{[n,i]}(C(X^{i}_n))$, $X^{i}_n$ are $[0,1]$, $k_n$, and $[n,i]$ are positive integers. Suppose that $A$ has the ideal property: each closed two-sided ideal of $A$ is generated by the projections inside the ideal, as a closed two-sided ideal. In this article, we give a complete classification of AI algebras with the ideal property.

Keywords:AI algebras, K-group, tracial state, ideal property, classification
Categories:46L35, 19K14, 46L05, 46L08

7. CJM 2011 (vol 63 pp. 500)

Dadarlat, Marius; Elliott, George A.; Niu, Zhuang
One-Parameter Continuous Fields of Kirchberg Algebras. II
Parallel to the first two authors' earlier classification of separable, unita one-parameter, continuous fields of Kirchberg algebras with torsion free $\mathrm{K}$-groups supported in one dimension, one-parameter, separable, uni continuous fields of AF-algebras are classified by their ordered $\mathrm{K}_0$-sheaves. Effros-Handelman-Shen type theorems are pr separable unital one-parameter continuous fields of AF-algebras and Kirchberg algebras.

Keywords:continuous fields of C$^*$-algebras, $\mathrm{K}_0$-presheaves, Effros--Handeen type theorem
Category:46L35

8. CJM 2008 (vol 60 pp. 703)

Toms, Andrew S.; Winter, Wilhelm
$\mathcal{Z}$-Stable ASH Algebras
The Jiang--Su algebra $\mathcal{Z}$ has come to prominence in the classification program for nuclear $C^*$-algebras of late, due primarily to the fact that Elliott's classification conjecture in its strongest form predicts that all simple, separable, and nuclear $C^*$-algebras with unperforated $\mathrm{K}$-theory will absorb $\mathcal{Z}$ tensorially, i.e., will be $\mathcal{Z}$-stable. There exist counterexamples which suggest that the conjecture will only hold for simple, nuclear, separable and $\mathcal{Z}$-stable $C^*$-algebras. We prove that virtually all classes of nuclear $C^*$-algebras for which the Elliott conjecture has been confirmed so far consist of $\mathcal{Z}$-stable $C^*$-algebras. This follows in large part from the following result, also proved herein: separable and approximately divisible $C^*$-algebras are $\mathcal{Z}$-stable.

Keywords:nuclear $C^*$-algebras, K-theory, classification
Categories:46L85, 46L35

9. CJM 2008 (vol 60 pp. 189)

Lin, Huaxin
Furstenberg Transformations and Approximate Conjugacy
Let $\alpha$ and $\beta$ be two Furstenberg transformations on $2$-torus associated with irrational numbers $\theta_1,$ $\theta_2,$ integers $d_1, d_2$ and Lipschitz functions $f_1$ and $f_2$. It is shown that $\alpha$ and $\beta$ are approximately conjugate in a measure theoretical sense if (and only if) $\overline{\theta_1\pm \theta_2}=0$ in $\R/\Z.$ Closely related to the classification of simple amenable \CAs, it is shown that $\af$ and $\bt$ are approximately $K$-conjugate if (and only if) $\overline{\theta_1\pm \theta_2}=0$ in $\R/\Z$ and $|d_1|=|d_2|.$ This is also shown to be equivalent to the condition that the associated crossed product \CAs are isomorphic.

Keywords:Furstenberg transformations, approximate conjugacy
Categories:37A55, 46L35

10. CJM 2005 (vol 57 pp. 351)

Lin, Huaxin
Extensions by Simple $C^*$-Algebras: Quasidiagonal Extensions
Let $A$ be an amenable separable $C^*$-algebra and $B$ be a non-unital but $\sigma$-unital simple $C^*$-algebra with continuous scale. We show that two essential extensions $\tau_1$ and $\tau_2$ of $A$ by $B$ are approximately unitarily equivalent if and only if $$ [\tau_1]=[\tau_2] \text{ in } KL(A, M(B)/B). $$ If $A$ is assumed to satisfy the Universal Coefficient Theorem, there is a bijection from approximate unitary equivalence classes of the above mentioned extensions to $KL(A, M(B)/B)$. Using $KL(A, M(B)/B)$, we compute exactly when an essential extension is quasidiagonal. We show that quasidiagonal extensions may not be approximately trivial. We also study the approximately trivial extensions.

Keywords:Extensions, Simple $C^*$-algebras
Categories:46L05, 46L35

11. CJM 2002 (vol 54 pp. 138)

Razak, Shaloub
On the Classification of Simple Stably Projectionless $\C^*$-Algebras
It is shown that simple stably projectionless $\C^S*$-algebras which are inductive limits of certain specified building blocks with trivial $\K$-theory are classified by their cone of positive traces with distinguished subset. This is the first example of an isomorphism theorem verifying the conjecture of Elliott for a subclass of the stably projectionless algebras.

Categories:46L35, 46L05

12. CJM 2001 (vol 53 pp. 161)

Lin, Huaxin
Classification of Simple Tracially AF $C^*$-Algebras
We prove that pre-classifiable (see 3.1) simple nuclear tracially AF \CA s (TAF) are classified by their $K$-theory. As a consequence all simple, locally AH and TAF \CA s are in fact AH algebras (it is known that there are locally AH algebras that are not AH). We also prove the following Rationalization Theorem. Let $A$ and $B$ be two unital separable nuclear simple TAF \CA s with unique normalized traces satisfying the Universal Coefficient Theorem. If $A$ and $B$ have the same (ordered and scaled) $K$-theory and $K_0 (A)_+$ is locally finitely generated, then $A \otimes Q \cong B \otimes Q$, where $Q$ is the UHF-algebra with the rational $K_0$. Classification results (with restriction on $K_0$-theory) for the above \CA s are also obtained. For example, we show that, if $A$ and $B$ are unital nuclear separable simple TAF \CA s with the unique normalized trace satisfying the UCT and with $K_1(A) = K_1(B)$, and $A$ and $B$ have the same rational (scaled ordered) $K_0$, then $A \cong B$. Similar results are also obtained for some cases in which $K_0$ is non-divisible such as $K_0(A) = \mathbf{Z} [1/2]$.

Categories:46L05, 46L35

13. CJM 2001 (vol 53 pp. 51)

Dean, Andrew
A Continuous Field of Projectionless $C^*$-Algebras
We use some results about stable relations to show that some of the simple, stable, projectionless crossed products of $O_2$ by $\bR$ considered by Kishimoto and Kumjian are inductive limits of type I $C^*$-algebras. The type I $C^*$-algebras that arise are pullbacks of finite direct sums of matrix algebras over the continuous functions on the unit interval by finite dimensional $C^*$-algebras.

Categories:46L35, 46L57

14. CJM 2000 (vol 52 pp. 1164)

Elliott, George A.; Villadsen, Jesper
Perforated Ordered $\K_0$-Groups
A simple $\C^*$-algebra is constructed for which the Murray-von Neumann equivalence classes of projections, with the usual addition---induced by addition of orthogonal projections---form the additive semi-group $$ \{0,2,3,\dots\}. $$ (This is a particularly simple instance of the phenomenon of perforation of the ordered $\K_0$-group, which has long been known in the commutative case---for instance, in the case of the four-sphere---and was recently observed by the second author in the case of a simple $\C^*$-algebra.)

Categories:46L35, 46L80

15. CJM 1997 (vol 49 pp. 963)

Lin, Huaxin
Homomorphisms from $C(X)$ into $C^*$-algebras
Let $A$ be a simple $C^*$-algebra with real rank zero, stable rank one and weakly unperforated $K_0(A)$ of countable rank. We show that a monomorphism $\phi\colon C(S^2) \to A$ can be approximated pointwise by homomorphisms from $C(S^2)$ into $A$ with finite dimensional range if and only if certain index vanishes. In particular, we show that every homomorphism $\phi$ from $C(S^2)$ into a UHF-algebra can be approximated pointwise by homomorphisms from $C(S^2)$ into the UHF-algebra with finite dimensional range. As an application, we show that if $A$ is a simple $C^*$-algebra of real rank zero and is an inductive limit of matrices over $C(S^2)$ then $A$ is an AF-algebra. Similar results for tori are also obtained. Classification of ${\bf Hom}\bigl(C(X),A\bigr)$ for lower dimensional spaces is also studied.

Keywords:Homomorphism of $C(S^2)$, approximation, real, rank zero, classification
Categories:46L05, 46L80, 46L35

© Canadian Mathematical Society, 2014 : https://cms.math.ca/