Vector-valued invariant means revisited once again

TL;DR

This paper revisits the concept of vector-valued invariant means in Banach spaces, correcting previous proofs and exploring their relation to complemented spaces in the second dual, with implications for amenable semigroups.

Contribution

It provides a corrected proof of the characterization of Banach spaces complemented in the second dual via invariant means and explores invariant means in universally separably injective spaces.

Findings

Corrected proof of the characterization of complemented Banach spaces

Universal separably injective spaces admit invariant means for countable amenable semigroups

Countable amenable semigroups are insufficient to characterize second dual complementability

Abstract

Banach spaces that are complemented in the second dual are characterised precisely as those spaces $X$ which enjoy the property that for every amenable semigroup $S$ there exists an $X$-valued analogue of an invariant mean defined on the Banach space of all bounded $X$-valued functions on $S$. This was first observed by Bustos Domecq (J. Math. Anal. Appl., 2002), however the original proof was slightly flawed as remarked by Lipecki. The primary aim of this note is to present a corrected version of the proof. We also demonstrate that universally separably injective spaces always admit invariant means with respect to countable amenable semigroups, thus such semigroups are not rich enough to capture complementation in the second dual as spaces falling into this class need not be complemented in the second dual.