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1. CMB 2011 (vol 56 pp. 424)
Convergent Sequences in Discrete Groups We prove that a finitely generated group contains a
sequence of non-trivial elements that converge to the identity in
every compact homomorphic image if and only if the group is not
virtually abelian. As a consequence of the methods used, we show that a finitely generated
group satisfies Chu duality if and only if it is virtually abelian.
Keywords:Chu duality, Bohr topology Category:54H11 |
2. 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 |