Representations of minimally almost periodic groups

Abstract

For any group G, we introduce the subset S(G) of elements g which are conjugate to g 2 , g 3 , g 4 ,... for some positive integer k. We show that, for any bounded representation w of G and any g in S(G), either w(g) = 1 or the spectrum of w(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.