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Proceedings of the Edinburgh Mathematical Society, vol.40, no.2, pp.261-266, 1997 (SCI-Expanded)
Let G be a locally compact abelian group, M0(G) be a closed regular subalgebra of the convolution measure algebra M(G) which contains the group algebra L1(G) and ω : M0(G) → B be a continuous homomorphism of M0(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 σB(ω(μ)) = μ̂(sp(ω)) holds for each mu; ∈ M0(G), where sp(ω) denotes the Arveson spectrum of ω, σB(.) the usual spectrum in B, μ̂ the Fourier-Stieltjes transform of μ.