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Megrelishvili defines \emph{light groups} of isomorphisms of a Banach spaceas the groups on which the Weak and Strong Operator Topologies coincide, andproves that every bounded group of isomorphisms of Banach spaces with the Pointof Continuity Property (PCP) is light. We investigate this concept forisomorphism groups $G$ of classical Banach spaces $X$ without the PCP,specially isometry groups, and relate it to the existence of $G$-invariant LURor strictly convex renormings of $X$.

Light groups of isomorphisms of Banach spaces and invariant LUR renormings