A note on Mackey topologies on Banach spaces

• Published in: Journal of mathematical analysis and applications 2017-01, Vol.445 (1), p.944-952
• Main Authors: Guirao, A.J., Montesinos, V., Zizler, V.
• Format: Article
• Language: English
• Subjects: Banach space; Completeness; Mackey topology; Mazur space; Norming subspace; Weak compactness
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2016.08.030

There is a maybe unexpected connection between three apparently unrelated notions concerning a given w⁎-dense subspace Y of the dual X⁎ of a Banach space X: (i) The norming character of Y, (ii) the fact that (Y,w⁎) has the Mazur property, and (iii) the completeness of the Mackey topology μ(X,Y), i.e., the topology on X of the uniform convergence on the family of all absolutely convex w⁎-compact subsets of Y. To clarify these connections is the purpose of this note. The starting point was a question raised by M. Kunze and W. Arendt and the answer provided by J. Bonet and B. Cascales. We fully characterize μ(X,Y)-completeness or its failure in the case of Banach spaces X with a w⁎-angelic dual unit ball—in particular, separable Banach spaces or, more generally, weakly compactly generated ones—by using the norming or, alternatively, the Mazur character of Y. We characterize the class of spaces where the original Kunze–Arendt question has always a positive answer. Some other applications are also provided.