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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 2011 (vol 64 pp. 705)
Pure Infiniteness of the Crossed Product of an AH-Algebra by an Endomorphism It is shown that simplicity of the crossed product of
a unital AH-algebra with slow dimension growth by an endomorphism
implies that the algebra is also purely infinite, provided only that
the endomorphism leaves no trace state invariant and takes the unit
to a full projection.
Keywords:purely infinite $C^*$-algebras, crossed products Category:46-xx |
3. 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 |
4. CJM 2007 (vol 59 pp. 343)
Weak Semiprojectivity in Purely Infinite Simple $C^*$-Algebras Let $A$ be a separable amenable purely infinite simple \CA which
satisfies the Universal Coefficient Theorem. We prove that $A$ is
weakly semiprojective if and only if $K_i(A)$ is a countable
direct sum of finitely generated groups ($i=0,1$). Therefore, if
$A$ is such a \CA, for any $\ep>0$ and any finite subset ${\mathcal
F}\subset A$ there exist $\dt>0$ and a finite subset ${\mathcal
G}\subset A$ satisfying the following: for any contractive
positive linear map $L: A\to B$ (for any \CA $B$) with $
\|L(ab)-L(a)L(b)\|<\dt$ for $a, b\in {\mathcal G}$
there exists a homomorphism $h\from A\to B$ such that
$ \|h(a)-L(a)\|<\ep$ for $a\in {\mathcal F}$.
Keywords:weakly semiprojective, purely infinite simple $C^*$-algebras Categories:46L05, 46L80 |
5. CJM 2006 (vol 58 pp. 1268)
Gauge-Invariant Ideals in the $C^*$-Algebras of Finitely Aligned Higher-Rank Graphs We produce a complete description of the lattice of gauge-invariant
ideals in $C^*(\Lambda)$ for a finitely aligned $k$-graph
$\Lambda$. We provide a condition on $\Lambda$ under which every ideal
is gauge-invariant. We give conditions on $\Lambda$ under which
$C^*(\Lambda)$ satisfies the hypotheses of the Kirchberg--Phillips
classification theorem.
Keywords:Graphs as categories, graph algebra, $C^*$-algebra Category:46L05 |
6. 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 |
7. CJM 2001 (vol 53 pp. 809)
Asymptotic $K$-Theory for Groups Acting on $\tA_2$ Buildings Let $\Gamma$ be a torsion free lattice in $G=\PGL(3, \mathbb{F})$ where
$\mathbb{F}$ is a nonarchimedean local field. Then $\Gamma$ acts freely
on the affine Bruhat-Tits building $\mathcal{B}$ of $G$ and there is an
induced action on the boundary $\Omega$ of $\mathcal{B}$. The crossed
product $C^*$-algebra $\mathcal{A}(\Gamma)=C(\Omega) \rtimes \Gamma$
depends only on $\Gamma$ and is classified by its $K$-theory. This article
shows how to compute the $K$-theory of $\mathcal{A}(\Gamma)$ and of the
larger class of rank two Cuntz-Krieger algebras.
Keywords:$K$-theory, $C^*$-algebra, affine building Categories:46L80, 51E24 |