Light groups of isomorphisms of Banach spaces and invariant LUR renormings

TL;DR

This paper explores the concept of light groups of isomorphisms in Banach spaces, especially those without the Point of Continuity Property, and examines their relation to invariant LUR or strictly convex renormings.

Contribution

It extends the study of light groups to classical Banach spaces without PCP and investigates their connection to invariant LUR and strictly convex renormings.

Findings

Light groups can be characterized for classical Banach spaces without PCP.

Existence of G-invariant LUR renormings is linked to the properties of the isomorphism groups.

Relations between isometry groups and invariant convexity properties are established.

Abstract

Megrelishvili defines \emph{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 $X$.