Operator topologies and reflexive representability (Q2762071)

For a Banach space \(X\), a subgroup \(G\) of \(GL(X)\) is said to be light if the weak and strong operator topologies coincide on \(G\). The Banach space \(X\) is called bound-fragmented (in short, BF) if every bounded subset of \(X\) is fragmented, and it is said to have the point of continuity property (PCP) if each bounded weakly closed subset \(C\) of \(X\) admits a point of continuity of the identity map \((C\), weak)\(\to(C\), norm). It turns out that these properties, BF and PCP are equivalent for every Banach space. The author proves that every bounded subgroup \(G\) of \(GL(X)\) is light for a Banach space \(X\) with the property PCP.NEWLINENEWLINEFor the entire collection see [Zbl 0970.00016].