Octahedrality in Lipschitz-free Banach spaces

Abstract

The aim of this note is to study octahedrality in vector valued Lipschitz-free Banach spaces on a metric space under topological hypotheses on it by analysing the weak-star strong diameter two property in Lipschitz functions spaces. Also, we show an example which proves that our results are optimal and that octahedrality in vectorvalued Lipschitz-free Banach spaces actually relies on the underlying metric space as well as on the Banach one.2010 Mathematics Subject Classification. Primary 46B20; Secondary 46B22, 52A10.

On the Dynamics of Lipschitz Operators

“…Next, we wish to highlight that in the literature on vector-valued Lipschitz free spaces (e.g. [16,20]), the notation F(M, C) refers to the projective tensor product F(M, R) ⊗ π C, where both factors are seen as real Banach spaces. This approach is motivated by the following R-linear isometric identifications:…”

On the Lipschitz Decomposition Problem in Ordered Banach Spaces and Its Connections to Other Branches of Mathematics

“…Vector-valued Lipschitz spaces has seen much recent developments. Of interest here is the observation by Guerrero, Lopez-Perez and Ruedo Zoca that, as in the scalar case, with X a Banach space, the space Lip 0 (M, X * ) is always a dual Banach space, having a vector-valued version of the Free Lipschitz space as predual [17].…”

“…It has long been known that scalar-valued Lipschitz function spaces are dual Banach spaces, with the Free Lipschitz space (also called the Arens-Eels space) as a predual [33]. Very recently in [17], dual-Banach-space valued Lipschitz function spaces were also shown to be dual Banach spaces, and furthermore, having a projective tensor product (with the Free Lipschitz space as tensor factor) as predual.…”