Expand all Collapse all | Results 1 - 14 of 14 |
1. 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 |
2. 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 |
3. 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 |
4. 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 |
5. 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 |
6. 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 |
7. 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 |
8. 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 |
9. 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 |
10. 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 |
11. 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 |
12. 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 |
13. 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 |
14. 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 |