1-complemented subspaces of Banach spaces of universal disposition

TL;DR

This paper unifies various notions of partial injectivity in Banach spaces and characterizes 1-complemented subspaces of universal disposition spaces as exactly the spaces with a specific injectivity property.

Contribution

It introduces a unified framework for partial injectivity and extends the concept of universal disposition to this setting, providing a complete characterization of 1-complemented subspaces.

Findings

Unified all notions of partial injectivity in Banach spaces.

Extended universal disposition to the $( ext{}eta)$-disposition setting.

Characterized 1-complemented subspaces as $( ext{}eta)_1$-injective.

Abstract

We first unify all notions of partial injectivity appearing in the literature ---(universal) separable injectivity, (universal) $\aleph$-injectivity --- in the notion of $(\alpha, \beta)$-injectivity ($(\alpha, \beta)_\lambda$-injectivity if the parameter $\lambda$ has to be specified). Then, extend the notion of space of universal disposition to space of universal $(\alpha, \beta)$-disposition. Finally, we characterize the $1$-complemented subspaces of spaces of universal $(\alpha, \beta)$-disposition as precisely the spaces $(\alpha, \beta)_1$-injective.