The group of automorphisms of $L_{\infty} $ is algebraically reflexive

Félix Cabello Sánchez¹

¹ Departamento de Matemáticas Universidad de Extremadura Avenida de Elvas 06071 Badajoz, Spain

Studia Mathematica, Tome 161 (2004) no. 1, pp. 19-32

Résumé

We study the reflexivity of the automorphism (and the isometry) group of the Banach algebras $L_\infty (\mu )$ for various measures $\mu $. We prove that if $\mu $ is a non-atomic $\sigma $-finite measure, then the automorphism group (or the isometry group) of $L_\infty (\mu )$ is [algebraically] reflexive if and only if $L_\infty (\mu )$ is $^*$-isomorphic to $L_\infty [0,1]$. For purely atomic measures, we show that the group of automorphisms (or isometries) of $\ell _\infty ({\mit \Gamma })$ is reflexive if and only if ${\mit \Gamma }$ has non-measurable cardinal. So, for most “practical" purposes, the automorphism group of $\ell _\infty ({\mit \Gamma })$ is reflexive.

Détail

DOI : 10.4064/sm161-1-2

Keywords: study reflexivity automorphism isometry group banach algebras infty various measures prove non atomic sigma finite measure automorphism group isometry group infty algebraically reflexive only infty * isomorphic infty purely atomic measures group automorphisms isometries ell infty mit gamma reflexive only mit gamma has non measurable cardinal practical purposes automorphism group ell infty mit gamma reflexive