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

Antonio M Peralta, Ignacio Villanueva, J D Maitland Wright, Kari Ylinen

Research output: Contribution to journalArticle

3 Citations (Scopus)
5 Downloads (Pure)

Abstract

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.
Original languageEnglish
Pages (from-to)1249-1267
Number of pages19
JournalProceedings of the Royal Society of Edinburgh A
Volume140
Issue number6
DOIs
Publication statusPublished - Dec 2010

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