Expand all Collapse all | Results 1 - 15 of 15 |
1. CJM Online first
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)
$K$-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 ErdÅs), 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)
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)
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)
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)
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)
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)
$\mathcal{Z}$-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)
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)
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)
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)
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)
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)
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)
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 |