Spectral Mapping Theorem for Representations of Measure Algebras

Abstract

Let G be a locally compact abelian group, M<inf>0</inf>(G) be a closed regular subalgebra of the convolution measure algebra M(G) which contains the group algebra L1(G) and ω : M<inf>0</inf>(G) → B be a continuous homomorphism of M<inf>0</inf>(G) into the unital Banach algebra B (possibly noncommutative) such that ω(L1(G)) is without order with respect to B in the sense that if for all b ∈ B, b.ω(L1(G)) = {0} implies b = 0. We prove that if sp(ω) is a synthesis set for L1(G) then the equality σ<inf>B</inf>(ω(μ)) = μ̂(sp(ω)) holds for each mu; ∈ M<inf>0</inf>(G), where sp(ω) denotes the Arveson spectrum of ω, σ<inf>B</inf>(.) the usual spectrum in B, μ̂ the Fourier-Stieltjes transform of μ.