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 bar right arrow parallel to Sx parallel to for bounded linear maps S from X into Hilbert spaces. The w-right topology for X rho(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.