Group actions on twisted sums of Banach spaces

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 [30]. Consequences are presented in the cases of certain non-unitarizable triangular representations of the group F ∞ on the Hilbert space, of the compatibility of complex structures on twisted sums, as well as of bounded actions on the interpolation scale of L p -spaces. As a new fundamental example, the isometry group of L p (0, 1), p = 2 is shown to extend as an isometry group acting on the associated Kalton-Peck space Z p [23]. 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 for the group F ∞ as well as for the space c 0 .

Quasilinear duality and inversion in Banach spaces

“…The role of the ambient space was considered in [10]. There it was shown that given two quasilinear maps Φ, Ψ : X Y , the former with ambient space A and the second with ambient space B then one can consider that both maps have a certain common space C as ambient space.…”

“…The notions of domain and range of centralizer-like mappings Ω : X Y have been considered in [7][8][9][10].…”

“…In this paper, we present a unified approach to the inversion and duality phenomena for centralizers and differential maps generated by interpolation processes, as studied in [7][8][9][10]. While duality is a standard topic, inversion emerged through results from several papers:…”