Group actions on twisted sums of Banach spaces

TL;DR

This paper investigates how group actions influence the structure of twisted sums of Banach spaces, characterizing these actions via quasilinear maps and exploring their properties under certain conditions.

Contribution

It introduces the notions of $G$-centralizer and $G$-equivariant map, providing characterizations and optimal results for bounded group actions on twisted sums of Banach spaces.

Findings

Bounded actions are generated by compatible families of operators under certain conditions.

Compatible quasilinear maps are linear perturbations of $G$-centralizers.

$G$-centralizers are bounded perturbations of $G$-equivariant maps under specific assumptions.

Abstract

We study bounded actions of groups and semigroups $G$ on exact sequences of Banach spaces from the point of view of quasilinear maps, characterize the actions on the twisted sum space by commutator estimates and introduce the associated notions of $G$-centralizer and $G$-equivariant map. We will show that when (A) $G$ is an amenable group and (U) the target space is complemented in its bidual by a $G$-equivariant projection, then uniformly bounded compatible families of operators generate bounded actions on the twisted sum space; that compatible quasilinear maps are linear perturbations of $G$-centralizers; and that, under (A) and (U), $G$-centralizers are bounded perturbations of $G$-equivariant maps. The previous results are optimal. Several examples and counterexamples are presented involving the action of the isometry group of $L_p(0,1), p\neq 2$ on the Kalton-Peck space $Z_p$, certain non-unitarizable triangular representations of the free group $F_\infty$ on the Hilbert space, the compatibility of complex structures on twisted sums, or bounded actions on the interpolation scale of $L_p$-spaces. In the last section we consider the category of $G$-Banach spaces and study its exact sequences, showing that, under (A) and (U), $G$-splitting and usual splitting coincide.