Amenability of algebras of approximable operators

TL;DR

This paper establishes a precise criterion for when the algebra of approximable operators on a Banach space is amenable, linking it to operator factorization and providing new methods to analyze specific spaces.

Contribution

It introduces a necessary and sufficient condition for amenability of the algebra of approximable operators and develops techniques to assess this property for various Banach spaces.

Findings

Characterization of amenability in terms of operator factorization

Non-amenability of the algebra on Tsirelson's space

Enhanced methods for analyzing algebraic amenability

Abstract

We give a necessary and sufficient condition for amenability of the Banach algebra of approximable operators on a Banach space. We further investigate the relationship between amenability of this algebra and factorization of operators, strengthening known results and developing new techniques to determine whether or not a given Banach space carries an amenable algebra of approximable operators. Using these techniques, we are able to show, among other things, the non-amenability of the algebra of approximable operators on Tsirelson's space.