Characterizations of weakly $\mathcal{K}$-analytic and Va\v{s}\'ak spaces using projectional skeletons and separable PRI

TL;DR

This paper characterizes Vašák and weakly fer spaces using projectional skeletons and SPRI, addressing recent open problems and providing new insights into WCG spaces and their subspaces.

Contribution

It introduces new characterizations of Vašák and weakly fer spaces via projectional skeletons and SPRI, solving a recent challenge and extending understanding of WCG spaces.

Findings

Characterizations of Vašák and weakly fer spaces using projectional skeletons.

Existence of a common projectional skeleton for countably many skeletons inducing the same set.

New insights into the structure of WCG spaces and their subspaces.

Abstract

We find characterizations of Va\v{s}\'ak spaces and weakly $\mathcal{K}$-analytic spaces using the notions of separable projectional resolution of the identity (SPRI) and of projectional skeleton. This in particular addresses a recent challenge suggested by M. Fabian and V. Montesinos in \cite{FM18}. Our method of proof also gives similar characterizations of WCG spaces and their subspaces (some aspects of which were known, some are new). Moreover we show that for countably many projectional skeletons $\{\mathfrak{s}_n: n \in \omega\}$ on a Banach space inducing the same set, there exists a projectional skeleton on the space (indexed by ranges of the corresponding projections) which is isomorphic to a subskeleton of each $\mathfrak{s}_n$, $n \in \omega$.