Spectra of independent power measures

Abstract

1. Introduction. We are concerned with measures µ, in the measure algebra M(G) of a locally compact Abelian group G, which have independent (mutually singular) powers. In (6), Williamson showed that the spectrum, σ(µ) of a Hermitian independent power measure µ satisfying ∥µ∥n=∥µn∥ for positive integer n, contains an infinity of points on the real axis. He conjectured that, in fact, σ(µ) is the disc {λ:|λ|≤∥µ∥}. Taylor (5), has recently proved that, in the case G = R, any positive continuous independent power µ has σ(µ) = {λ:|λ|≤∥µ∥}. His methods depend on his deep and beautiful theory of critical points. Here we verify Williamson's conjecture, give an elementary proof of Taylor's result, and give a simple characterization of the class of LCA groups for which the natural generalization is valid.