Weakly compact operators and the strong∗ topology for a Banach space

The strong∗ topology s∗(X) of a Banach space X is defined as the locally convex topology generated by the seminorms x → ‖Sx‖ for bounded linear maps S from X into Hilbert spaces. The w-right topology for X, ρ(X), is a stronger locally convex topology, which may be analogously characterized by taking reflexive Banach spaces in place of Hilbert spaces. For any Banach space Y , a linear map T : X → Y is known to be weakly compact precisely when T is continuous from the w-right topology to the norm topology of Y . The main results deal with conditions for, and consequences of, the coincidence of these two topologies on norm bounded sets. A large class of Banach spaces, including all C∗-algebras and, more generally, all JB∗-triples, exhibit this behaviour.