Group Actions on Twisted Sums of Banach Spaces

We study bounded actions of groups and semigroups G on exact sequences of Banach spaces from the point of view of (generalized) 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 not equal 2 on the Kalton-Peck space Z(p), certain non-unitarizable triangular representations of the free group F-infinity 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 penultimate 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. The purpose of the final section is to present some applications, showing that several previous result are optimal and to suggest further open lines of research.