(Non-)amenability of B(E)

TL;DR

This paper surveys the history of amenability in Banach algebras of bounded operators on infinite-dimensional spaces and presents a proof that $B(\, ext{ell}^p)$ is non-amenable for all p in [1, infinity].

Contribution

It provides a comprehensive survey of previous results and introduces a new proof establishing non-amenability of $B(\, ext{ell}^p)$ for all p in [1, infinity].

Findings

$B(\, ext{ell}^p)$ is non-amenable for p=1,2,∞.

Recent results show $B(E)$ can be amenable for some infinite-dimensional E.

The paper extends non-amenability to all p in [1,∞] for $B(\, ext{ell}^p)$.

Abstract

In 1972, the late B. E. Johnson introduced the notion of an amenable Banach algebra and asked whether the Banach algebra $B(E)$ of all bounded linear operators on a Banach space $E$ could ever be amenable if $\dim E = \infty$. Somewhat surprisingly, this question was answered positively only very recently as a by-product of the Argyros--Haydon result that solves the "scalar plus compact problem": there is an infinite-dimensional Banach space $E$, the dual of which is $\ell^1$, such that $B(E) = K(E)+ \mathbb{C} \id_E$. Still, $B(\ell^2)$ is not amenable, and in the past decade, $ B(\ell^p)$ was found to be non-amenable for $p=1,2,\infty$ thanks to the work of C. J. Read, G. Pisier, and N. Ozawa. We survey those results, and then--based on joint work with M. Daws--outline a proof that establishes the non-amenability of $B(\ell^p)$ for all $p \in [1,\infty]$.