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.

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