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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 μ ∘ x, μ ∈ M(G), x ∈ X. We show that if X is an essential L¹(G)-module, then $\sigma \left(T_{\mu }\right)=\overline{\mu ̂\left(sp\right(X\left)\right)}$ for each measure μ in reg(M(G)), where $T_{\mu }$ denotes the operator in B(X) defined by $T_{\mu }x=\mu \circ x$, σ(·) the usual spectrum in B(X), sp(X) the hull in L¹(G) of the ideal $I_X=f\in L¹\left(G\right)|T_f=0$, μ̂ the Fourier-Stieltjes transform of μ, and reg(M(G)) the largest closed regular subalgebra of M(G); reg(M(G)) contains all...