This is an announcement for the paper “Group actions on twisted sums of Banach spaces” by Jesús M.F. Castillohttps://arxiv.org/search/math?searchtype=author&query=Castillo%2C+J+M+F, Valentin Ferenczihttps://arxiv.org/search/math?searchtype=author&query=Ferenczi%2C+V.
Abstract: We study bounded actions of groups and semigroups on exact sequences of Banach spaces, characterizing different type of actions in terms of commutator estimates satisfied by the quasi-linear map associated to the exact sequence. As a special and important case, actions on interpolation scales are related to actions on the exact sequence induced by the scale through the Rochberg-Weiss theory. Consequences are presented in the cases of certain non-unitarizable triangular representations of the free group on the Hilbert space, of the compatibility of complex structures on twisted sums, as well as of bounded actions on the interpolation scale of Lp-spaces. As a new fundamental example, the isometry group of Lp(0,1), p different from 2, is shown to extend as an isometry group acting on the associated Kalton-Peck space Zp. Finally we define the concept of G-splitting for exact sequences admitting the action of a semigroup G, and give criteria and examples to relate G-splitting and usual splitting of exact sequences: while both are equivalent for amenable groups and, for example, reflexive spaces, counterexamples are provided where one of these hypotheses is not satisfied.