Spectral mapping theorem for representations of measure algebras

Abstract

Let G be a locally compact abelian group, M-0(G) be a closed regular subalgebra of the convolution measure algebra M(G) which contains the group algebra L-1(G) and omega: M-0(G) --> B be a continuous homomorphism of M-0(G) into the unital Banach algebra B (possibly noncommutative) such that omega(L-1(G)) is without order with respect to B in the sense that if for all b is an element of B, b.omega(L-1(G)) = {O} implies b = 0. We prove that if sp(omega) is a synthesis see for L-1(G) then the equality sigma(B)(omega(mu)) = <(mu)over cap>(sp(omega)) holds for each mu is an element of M-0(G), where sp(omega) denotes the Arveson spectrum of omega, sigma(B)(.) the usual spectrum in B, <(mu)over cap> the Fourier-Stieltjes transform of mu.