The group of automorphisms of $L_{∞}$ is algebraically reflexive

Félix Cabello Sánchez

The group of automorphisms of

L_{\infty}

is algebraically reflexive

Félix Cabello Sánchez

Studia Mathematica, Tome 162 (2004), p. 19-32 / Harvested from The Polish Digital Mathematics Library

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Résumé

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

Publié le : 2004-01-01

Zbl 1057.46048

EUDML-ID : urn:eudml:doc:285240

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     title = {The group of automorphisms of $L\_{$\infty$}$ is algebraically reflexive},
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     volume = {162},
     year = {2004},
     pages = {19-32},
     zbl = {1057.46048},
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Félix Cabello Sánchez. The group of automorphisms of $L_{∞}$ is algebraically reflexive. Studia Mathematica, Tome 162 (2004) pp. 19-32. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm161-1-2/