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Representations of minimally almost periodic groups

Part of: Linear algebraic groups and related topics Locally compact groups and their algebras Selfadjoint operator algebras

Published online by Cambridge University Press:  09 April 2009

Alain Valette
Alain Valette
Affiliation:
Department of Mathematics CP 216, Université Libre de Bruxelles, Boulevard du Triomphe, B-1050 Brussels, Belgium
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Abstract

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For any group G, we introduce the subset S(G) of elements g which are conjugate to for some positive integer k. We show that, for any bounded representation π of G any g in S(G), either π(g) = 1 or the spectrum of π(g) is the full unit circle in C. As a corollary, S(G) is in the kernel of any homomorphism from G to the unitary group of a post-liminal C*-algebra with finite composition series.

Next, for a topological group G, we consider the subset of elements approximately conjugate to 1, and we prove that it is contained in the kernel of any uniformly continuous bounded representation of G, and of any strongly continuous unitary representation in a finite von Neumann algebra.

We apply these results to prove triviality for a number of representations of isotropic simple algebraic groups defined over various fields.

MSC classification

Secondary: 22D12: Other representations of locally compact groups 20G15: Linear algebraic groups over arbitrary fields 46L05: General theory of $C^*$-algebras
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 41 , Issue 3 , December 1986 , pp. 366 - 375
Copyright
Copyright © Australian Mathematical Society 1986

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