Central Idempotent Measures on Unitary Groups

Abstract

Let G be a locally compact group and M(G) the space of finite regular Borel measures on G. If μ and v are in M(G), their convolution is defined byThus, if f is a continuous bounded function on G,μ is central if μ(Ex) = μ(xE) for all x ∈ G and all measurable sets E. μ is idempotent if μ * μ = μ.The idempotent measures for abelian groups have been classified by Cohen [1]. In this paper we will show that for a certain class of compact groups, containing the unitary groups, the central idempotents can be characterized. The method consists of showing that, in these cases, the central idempotents arise from idempotents on abelian groups and applying Cohen's result.