Tameness and Rosenthal type locally convex spaces

TL;DR

This paper introduces the class of tame locally convex spaces, extending Rosenthal's dichotomy to a broader setting and providing characterizations, properties, and operator factorization results related to tameness.

Contribution

It generalizes Rosenthal's dichotomy to locally convex spaces, characterizes tameness via extreme points and sequences, and shows tame operators factor through tame Banach spaces.

Findings

Tame lcs spaces are characterized by properties of their duals.

Tame spaces exclude sequences equivalent to generalized l^1 sequences.

Operators between tame spaces and Banach spaces factor through tame Banach spaces.

Abstract

Motivated by Rosenthal's famous $l^1$-dichotomy in Banach spaces, Haydon's theorem, and additionally by recent works on tame dynamical systems, we introduce the class of tame locally convex spaces. This is a natural locally convex analogue of Rosenthal Banach spaces (for which any bounded sequence contains a weak Cauchy subsequence). Our approach is based on a bornology of tame subsets which in turn is closely related to eventual fragmentability. This leads, among others, to the following results: $\bullet$ extending Haydon's characterization of Rosenthal Banach spaces, by showing that a lcs $E$ is tame iff every weak-star compact, equicontinuous convex subset of $E^{*}$ is the strong closed convex hull of its extreme points iff $\overline{\rm{co\,}}^{w^{*}}(K) = \overline{\rm{co\,}}(K)$ for every weak-star compact equicontinuous subset $K$ of $E^{*}$; $\bullet$ $E$ is tame iff there is no bounded sequence equivalent to the generalized $l^{1}$-sequence; $\bullet$ strengthening some results of W.M. Ruess about Rosenthal's dichotomy; $\bullet$ applying the Davis-Figiel-Johnson-Pelczy\'nski (DFJP) technique one may show that every tame operator $T \colon E \to F$ between a lcs $E$ and a Banach space $F$ can be factored through a tame (i.e., Rosenthal) Banach space.