A spectral mapping theorem for Banach modules

H. Seferoğlu¹

¹ Department of Mathematics Faculty of Art and Science Yüzüncü Yil University 65080 Van, Turkey

Studia Mathematica, Tome 156 (2003) no. 2, pp. 99-103

Résumé

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.

Détail

DOI : 10.4064/sm156-2-1

Keywords: locally compact abelian group convolution measure algebra banach module under module multiplication circ essential module sigma overline widehat mathop each measure mathop reg nolimits where denotes operator defined circ sigma cdot usual spectrum mathop hull ideal mid widehat fourier stieltjes transform mathop reg nolimits largest closed regular subalgebra mathop reg nolimits contains absolutely continuous measures discrete measures