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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 $\mu \circ x$, $\mu \in M(G)$, $x \in X$. We show that if $X$ is an essential $L^{1}(G)$-module, then $\sigma (T_{\mu })=\overline {\widehat {\mu }(\mathop{\rm sp}(X))} $ for each measure $\mu $ in $\mathop{\rm reg}\nolimits(M(G))$, where $T_{\mu }$ denotes the operator in $B(X)$ defined by $T_{\mu }x = \mu \circ x$, $\sigma ( \cdot )$ the usual spectrum in $B(X)$, $\mathop{\rm sp}(X)$ the hull in $L^{1}(G)$ of the ideal $I_{X }= \{f \in L^{1}(G) \mid T_{f }= 0\}$, $\widehat {\mu }$ the Fourier–Stieltjes transform of $\mu$, and $\mathop{\rm reg}\nolimits(M(G))$ the largest closed regular subalgebra of $M(G)$; $\mathop{\rm reg}\nolimits(M(G))$ contains all the absolutely continuous measures and discrete measures.