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1. CMB Online first
Simplicity of Partial Skew Group Rings of Abelian Groups Let $A$ be a ring with local units, $E$ a set of local units for $A$,
$G$ an abelian group and $\alpha$ a partial action of $G$ by ideals of
$A$ that contain local units.
We show that $A\star_{\alpha} G$ is simple if and only if $A$ is
$G$-simple and the center of the corner $e\delta_0 (A\star_{\alpha} G)
e \delta_0$ is a field for all $e\in E$. We apply the result to
characterize simplicity of partial skew group rings in two cases,
namely for partial skew group rings arising from partial actions by
clopen subsets of a compact set and partial actions on the set level.
Keywords:partial skew group rings, simple rings, partial actions, abelian groups Categories:16S35, 37B05 |
2. CMB 2012 (vol 56 pp. 709)
Universal Minimal Flows of Groups of Automorphisms of Uncountable Structures It is a well-known fact, that the greatest ambit for
a topological group $G$ is the Samuel compactification of $G$ with
respect to the right uniformity on $G.$ We apply the original
description by Samuel from 1948 to give a simple computation of the
universal minimal flow for groups of automorphisms of uncountable
structures using FraÃ¯ssÃ© theory and Ramsey theory. This work
generalizes some of the known results about countable structures.
Keywords:universal minimal flows, ultrafilter flows, Ramsey theory Categories:37B05, 03E02, 05D10, 22F50, 54H20 |
3. CMB 2011 (vol 55 pp. 297)
The Group $\operatorname{Aut}(\mu)$ is Roelcke Precompact Following a similar result of Uspenskij on the unitary group of a
separable Hilbert space, we show that, with respect to the lower (or
Roelcke) uniform structure, the Polish group $G=
\operatorname{Aut}(\mu)$ of automorphisms of an atomless standard
Borel probability space $(X,\mu)$ is precompact. We identify the
corresponding compactification as the space of Markov operators on
$L_2(\mu)$ and deduce that the algebra of right and left uniformly
continuous functions, the algebra of weakly almost periodic functions,
and the algebra of Hilbert functions on $G$, i.e., functions on
$G$ arising from unitary representations, all coincide. Again
following Uspenskij, we also conclude that $G$ is totally minimal.
Keywords:Roelcke precompact, unitary group, measure preserving transformations, Markov operators, weakly almost periodic functions Categories:54H11, 22A05, 37B05, 54H20 |
4. CMB 2007 (vol 50 pp. 418)
A Short Proof of Affability for Certain Cantor Minimal $\Z^2$-Systems We will show that any extension of a product of two Cantor minimal
$\Z$-systems is affable in the sense of Giordano, Putnam and Skau.
Category:37B05 |