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1. CJM Online first
Outer Partial Actions and Partial Skew Group Rings We extend the classicial notion of an outer action
$\alpha$ of a group $G$ on a unital ring $A$
to the case when $\alpha$ is a partial action
on ideals, all of which have local units.
We show that if $\alpha$ is an outer partial
action of an abelian group $G$,
then its associated partial skew group
ring $A \star_\alpha G$ is simple if and only if
$A$ is $G$-simple.
This result is applied to partial skew group rings associated with two different types of partial dynamical systems.
Keywords:outer action, partial action, minimality, topological dynamics, partial skew group ring, simplicity Categories:16W50, 37B05, 37B99, 54H15, 54H20 |
2. CJM 2014 (vol 67 pp. 330)
Hyperspace Dynamics of Generic Maps of the Cantor Space We study the hyperspace dynamics induced from generic continuous maps
and from generic homeomorphisms of the Cantor space, with emphasis on the
notions of Li-Yorke chaos, distributional chaos, topological entropy,
chain continuity, shadowing and recurrence.
Keywords:cantor space, continuous maps, homeomorphisms, hyperspace, dynamics Categories:37B99, 54H20, 54E52 |
3. 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 |
4. CJM 2002 (vol 54 pp. 1187)
On the Injectivity of $C^1$ Maps of the Real Plane Let $X\colon\mathbb{R}^2\to\mathbb{R}^2$ be a $C^1$ map. Denote by $\Spec(X)$ the set of
(complex) eigenvalues of $\DX_p$ when $p$ varies in $\mathbb{R}^2$. If there exists
$\epsilon >0$ such that $\Spec(X)\cap(-\epsilon,\epsilon)=\emptyset$, then
$X$ is injective. Some applications of this result to the real Keller Jacobian
conjecture are discussed.
Categories:34D05, 54H20, 58F10, 58F21 |
5. CJM 2001 (vol 53 pp. 325)
Ext and OrderExt Classes of Certain Automorphisms of $C^*$-Algebras Arising from Cantor Minimal Systems |
Ext and OrderExt Classes of Certain Automorphisms of $C^*$-Algebras Arising from Cantor Minimal Systems Giordano, Putnam and Skau showed that the transformation group
$C^*$-algebra arising from a Cantor minimal system is an $AT$-algebra,
and classified it by its $K$-theory. For approximately inner
automorphisms that preserve $C(X)$, we will determine their classes in
the Ext and OrderExt groups, and introduce a new invariant for the
closure of the topological full group. We will also prove that every
automorphism in the kernel of the homomorphism into the Ext group is
homotopic to an inner automorphism, which extends Kishimoto's result.
Categories:46L40, 46L80, 54H20 |