When is the Szlenk derivation of a dual unit ball another ball?

Speaker(s)

Tomasz Kochanek

Affiliation

UW

Language of the talk

English

Date

Dec. 4, 2024, 4:15 p.m.

Room

room 5050

Title in Polish

When is the Szlenk derivation of a dual unit ball another ball?

Seminar

Topology and Set Theory Seminar

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The Szlenk derivation of the dual unit ball of a Banach space is an analogue of the Cantor-Bendixson derivative, and the main ingredient in the definition of Szlenk index, a classical ordinal index with many applications in Banach space theory. In this talk, we are interested in characterizing the situation where all ɛ-Szlenk derivations of the unit ball of X* are balls. We prove that this is the case if X is separable and satisfies Kalton's property (M*). However, the converse implication is not true, as is witnessed by the dual of Baernstein's space which fails property (M*), but for which all ɛ-Szlenk derivation of the dual unit ball are balls with the same radius as for the separable Hilbert space. We will also derive estimates for the radii of enveloping balls of ɛ-Szlenk derivations for Tsirelson's space and the dual of Schlumprecht's space.

This is a joint work with Marek Miarka; preprint available at: https://arxiv.org/abs/2409.05516

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