Dual spaces of groups with precompact conjugacy classes

Abstract

We show that a second countable locally compact type I group with a compact invariant neighborhood of the identity is CCR, and has a Hausdorff dual if and only if its conjugacy classes are precompact. We obtain sharper results if the group is almost connected or has a fundamental system of invariant neighborhoods of the identity. Along the way we show that for a locally compact abelian group A A and a group B B of topological group automorphisms of A , A A, A has small B B invariant neighborhoods at 1 if and only if A ^ \hat A has precompact orbits under the dual actions of B B .