Invariant means on Abelian groups capture complementability of Banach spaces in their second duals

TL;DR

This paper characterizes when a Banach space is complemented in its bidual using invariant means on Abelian groups, providing a new criterion that improves previous results and answers an open question.

Contribution

It introduces a new invariant mean criterion for complementability of Banach spaces in their biduals, linking group invariance to functional analysis properties.

Findings

Characterizes complementability via invariant means on Abelian groups.

Establishes equivalence between invariant means and complementability.

Provides a new perspective on the structure of finite-dimensional subspaces.

Abstract

Let $X$ be a Banach space. Then $X$ is complemented in the bidual $X^{**}$ if and only if there exists an invariant mean $\ell_\infty(G, X)\to X$ with respect to a free Abelian group $G$ of rank equal to the cardinality of $X^{**}$, and this happens if and only if there exists an invariant mean with respect to the additive group of $X^{**}$. This improves upon previous results due to Bustos Domecq =and the second-named author, where certain idempotent semigroups of cardinality equal to the cardinality of $X^{**}$ were considered, and answers a question of J.M.F. Castillo (private communication). En route to the proof of the main result, we endow the family of all finite-dimensional subspaces of an infinite-dimensional vector space with a structure of a free commutative monoid with the property that the product of two subspaces contains the respective subspaces, which is possibly of interest in itself.