A Spectral Mapping Theorem for Banach Modules

Abstract

Let G be a locally compact abelian group, M(G) the convolution measure algebra, and X a Banach M(G)-module under the module multiplication muox, mu is an element of M(G), x is an element of X. We show that if X is an essential L-1 (G)-module, then sigma(T-mu) = (μ) over cap (sp(X)) for each measure mu in reg(M(G)), where T-mu denotes the operator in B(X) defined by T(mu)x = mu o x, sigma(.) the usual spectrum in B(X), sp(X) the hull in L-1(G) of the ideal I-X = {f is an element of L-1(G) \ T-f = 0}, the Fourier-Stieltjes transform of mu, and reg(M(G)) the largest closed regular subalgebra of M(G); reg(M(G)) contains all the absolutely continuous measures and discrete measures.