Group actions on twisted sums of Banach spaces
Jes\'us M.F. Castillo, Valentin Ferenczi

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 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 -centralizer and -equivariant map. We will show that when (A) is an amenable group and (U) the target space is complemented in its bidual by a -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 -centralizers; and that, under (A) and (U), -centralizers are bounded perturbations of -equivariant maps. The previous results are optimal. Several examples and counterexamples are presented involving the action of the isometry group of on the Kalton-Peck space ,…
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Taxonomy
TopicsAdvanced Banach Space Theory · Advanced Operator Algebra Research · Holomorphic and Operator Theory
