Rajchman measures on compact groups

Abstract

The aim of this paper is to characterize Radon measures whose coefficient operators have norms "vanishing at infinity" by the property of putting no mass on a certain class of sets. Rajchman [7] conjectured that such a class of sets exists for the circle group, grelder [9] defined so called W-sets with which he claimed to be able to prove Rajchman's assertion. The first to give a complete proof was Lyons. See Lyons [6] for the proof for locally compact Abelian groups. We extend this theorem from locally compact Abelian to compact non-Abelian groups. Let G be a compact group with dual object G. For each a e G let U~):G~B(H~), x-~U~)=(ul~)(x)) be a fixed continuous irreducible unitary representation of class a on the d:dimensional Hilbertspace H , = Hn with o.n.-basis {{I~)}, {I "~ = ~I ~) s.t. U Cn) = U ~'), # being the class of representations conjugate to a. The d~Z-dimensional space spanned by the coordinate functions ul~ ~ of the representation U <~) is denoted by M ~'~. For/~ in Jg(G), the set of complex Radon mesures on G, A, e B(H~) is the coefficient operator