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Abstract

In 1972, the late B. E. Johnson introduced the notion of an amenable Banach algebra and asked whether the Banach algebra ${\cal B}(E)$ 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 ${\cal B}(E) = {\cal K}(E)+ {\mathbb C}\, {\rm id}_E$. Still, ${\cal B}(\ell^2)$ is not amenable, and in the past decade, ${\cal B}(\ell^p)$ 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 ${\cal B}(\ell^p)$ for all $p \in [1,\infty]$.