Spectral Properties of Abelian C*-Algebras

Let be an Abelian unital C^*-algebra and let denote its Gelfand spectrum. We give some necessary and sufficient conditions for a nondegenerate representation of to be unitarily equivalent to a representation in which the elements of act multiplicatively, by their Gelfand transforms, on a space L²(,), where is a positive measure on the Baire sets of . We also compare these conditions with the multiplicity-free property of a representation.