TY - JOUR

T1 - Weakly compact operators and the strong* topology for a Banach space

AU - Peralta, Antonio M

AU - Villanueva, Ignacio

AU - Wright, J D Maitland

AU - Ylinen, Kari

PY - 2010/12

Y1 - 2010/12

N2 - 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.

AB - 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.

U2 - 10.1017/S0308210509001486

DO - 10.1017/S0308210509001486

M3 - Article

VL - 140

SP - 1249

EP - 1267

JO - Proceedings of the Royal Society of Edinburgh A

JF - Proceedings of the Royal Society of Edinburgh A

SN - 0308-2105

IS - 6

ER -