Light groups of isomorphisms of Banach spaces and invariant LUR renormings

Leandro Antunes , Valentin Ferenczi , Sophie Grivaux (1) , Christian Rosendal

1 LPP - Laboratoire Paul Painlevé - UMR 8524

Megrelishvili defines in [17] light groups of isomorphisms of a Banach space as the groups on which the Weak and Strong Operator Topologies coincide, and proves that every bounded group of isomorphisms of Banach spaces with the Point of Continuity Property (PCP) is light. We investigate this concept for isomorphism groups G of classical Banach spaces X without the PCP, specially isometry groups, and relate it to the existence of G-invariant LUR or strictly convex renormings of X.

Mots clés: groups of isomorphisms of Banach spaces; isometry groups; light groups; renormings of Banach spaces; LUR renormings; invariant renormings.

Domaines: Mathématiques [math]; Analyse fonctionnelle [math.FA]

Leandro Antunes, Valentin Ferenczi, Sophie Grivaux, Christian Rosendal. Light groups of isomorphisms of Banach spaces and invariant LUR renormings. Pacific Journal of Mathematics, 2019, 301 (1), pp.31-54. ⟨10.2140/pjm.2019.301.31⟩. ⟨hal-02140440⟩