Multiplicity-free representations of commutative C*-algebras and spectral properties

Abstract

Let A be a commutative unital C*-algebra and let S denote its Gelfand spectrum. We give some necessary and sufficient conditions for a nondegenerate representation of A to be unitarily equivalent to a multiplicative representation on a space L^2(S, m), where m is a positive measure on the Baire sets of S. We also compare these conditions with the multiplicity-free property of a representation.