Journal of Operator Theory

Volume 42, Issue 2, Fall 1999 pp. 245-267.

Algebras of multiplication operators in Banach function spaces

Authors: B. de Pagter (1), and W.J. Ricker (2)
Author institution: (1) Department of Mathematics & Informatics, Technical University Delft, 2600 AJ Delft, The Netherlands
(2) School of Mathematics, University of New South Wales, Sydney, 2052, Australia

Summary: Let $E$ be a Banach function space based on a Maharam measure $\mu$. For each $\varphi\in L^\infty(\mu)$, the linear operator $M_\varphi$ of multiplication by $\varphi$ is continuous on $E$. Let $\f{U}$ be a subalgebra of $L^\infty(\mu)$. We make a detailed study of the relationship between $\f{M}_E(\f{U})=\{M_\varphi : \varphi\in \f{U}\}$, the weak operator closed algebra $\ov{\f{M}_E(\f{U})^{\rm w}}$ it generates, the bicommutant algebra $\f{M}_E(\f{U})^{\rm cc}$, and the algebra $\f{M}_E(\ov{\f{U}^*})$, where $\ov{\f{U}^*}$ is the weak-$*$ closure of $\f{U}$ in $L^\infty(\mu)$. When $E$ is fully symmetric it is shown that $$ \f{M}_E (\f{U}) \subseteq \ov{\f{M}_E (\f{U})^{\rm w}} \subseteq \f{M}_E (\f{U})^{\cc} \subseteq \f{M}_E (\ov{\f{U}^*}) \subseteq \f{M}_E (L^\infty (\mu)). $$ The inclusion $\f{M}_E (\f{U})^{\cc} \subseteq \f{M}_E (\ov{\f{U}^*})$ may fail if $E$ is not fully symmetric.

Keywords: Fully symmetric Banach function space, multiplication operator, bicommutant

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