Representations of minimally almost periodic groups

Abstract

AbstractFor any groupG, we introduce the subsetS(G) of elementsgwhich are conjugate tofor some positive integerk. We show that, for any bounded representation π ofGanyginS(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 fromGto the unitary group of a post-liminalC*-algebra with finite composition series.Next, for a topological groupG, 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 ofG, 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.