Banach spaces with a large semigroup of contractive automorphisms

Abstract

Transitivity, almost transitivity, and convex-transitivity are classical notions in the isometric theory of Banach spaces, all of them related to Mazur's problem asking whether transitive separable Banach spaces are Hilbert spaces [24,33]. In this paper we study ‘near’ versions of the above notions. Roughly speaking, ‘nearness’ means that contractive automorphisms play the role played by surjective linear isometries in the corresponding classical notion. We also consider asymptotic transitivity as defined in [35], which is implied by almost transitivity and implies near almost transitivity, and introduce the corresponding notion of asymptotic convex-transitivity. We show that nearly convex-transitive (respectively, asymptotically convex-transitive) Banach spaces are nearly almost-transitive (respectively, asymptotically transitive), uniformly smooth, and uniformly convex, as soon as they be Asplund spaces or have the Radon-Nikodým property. We also deal with a property stronger than the near transitivity, and weaker than the uniform micro-semitransitivity [18]. We prove that Banach spaces with this intermediate property are uniformly smooth and uniformly convex. Finally we show that nearly almost-transitive norm-unital complete normed (possibly non-associative) algebras are Hilbert spaces.