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.
Peralta, A. M., Villanueva, I., Wright, J. D. M., & Ylinen, K. (2010). Weakly compact operators and the strong* topology for a Banach space. Proceedings of the Royal Society of Edinburgh A, 140(6), 1249-1267. https://doi.org/10.1017/S0308210509001486