B(l^p) is never amenable

TL;DR

This paper proves that certain Banach algebras related to al^p spaces are not amenable, extending non-amenability results to a broad class of operators on al^p and L^p spaces.

Contribution

It establishes the non-amenability of al^p spaces' operator algebras, generalizing previous results and providing new insights into their algebraic structure.

Findings

al( abla^p) is not amenable for p  1,

al( abla^p) is not amenable for any infinite-dimensional al^p space

al( abla^p) is not amenable for al( abla^p) and al(L^p[0,1])

Abstract

We show that, if $E$ is a Banach space with a basis satisfying a certain condition, then the Banach algebra $\ell^\infty({\cal K}(\ell^2 \oplus E))$ is not amenable; in particular, this is true for $E = \ell^p$ with $p \in (1,\infty)$. As a consequence, $\ell^\infty({\cal K}(E))$ is not amenable for any infinite-dimensional ${\cal L}^p$-space. This, in turn, entails the non-amenability of ${\cal B}(\ell^p(E))$ for any ${\cal L}^p$-space $E$, so that, in particular, ${\cal B}(\ell^p)$ and ${\cal B}(L^p[0,1])$ are not amenable.