ON UNIVERSAL SPACES FOR THE CLASS OF BANACH SPACES WHOSE DUAL BALLS ARE UNIFORM EBERLEIN COMPACTS

Author(s)

Brech, Christina ^[1] ; Koszmider, Piotr ^[2]

Affiliation

^[1] Univ Sao Paulo, Inst Matemat & Estat, Dept Matemat, BR-05314970 Sao Paulo - Brazil

^[2] Tech Univ Lodz, Inst Math, PL-90924 Lodz - Poland

• Document type: Journal article
• Source: Proceedings of the American Mathematical Society; v. 141, n. 4, p. 1267-1280, APR 2013.
• Web of Science Citations: 3

Abstract

For kappa being the first uncountable cardinal omega(1) or kappa being the cardinality of the continuum c, we prove that it is consistent that there is no Banach space of density kappa in which it is possible to isomorphically embed every Banach space of the same density which has a uniformly Gateaux differentiable renorming or, equivalently, whose dual unit ball with the weak{*} topology is a subspace of a Hilbert space (a uniform Eberlein compact space). This complements a consequence of results of M. Bell and of M. Fabian, G. Godefroy, and V. Zizler which says that assuming the continuum hypothesis, there is a universal space for all Banach spaces of density kappa = c = omega(1) that have a uniformly Gateaux differentiable renorming. Our result implies, in particular, that beta N\textbackslash{}N may not map continuously onto a compact subset of a Hilbert space with the weak topology of density kappa = omega(1) or kappa = c and that a C(K) space for some uniform Eberlein compact space K may not embed isomorphically into l(infinity)/c(0). (AU)

• FAPESP's process: 10/12638-1 - Bases and universality in Banach spaces
• Grantee: Christina Brech
• Support Opportunities: Regular Research Grants