Non-amenability of BE - Mathematics > Functional AnalysisReport as inadecuate

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Abstract: In 1972, the late B. E. Johnson introduced the notion of an amenable Banachalgebra and asked whether the Banach algebra $BE$ of all bounded linearoperators on a Banach space $E$ could ever be amenable if $\dim E = \infty$.Somewhat surprisingly, this question was answered positively only very recentlyas a by-product of the Argyros-Haydon result that solves the -scalar pluscompact problem-: there is an infinite-dimensional Banach space $E$, the dualof which is $\ell^1$, such that $BE = KE+ \mathbb{C} \id E$. Still,$B\ell^2$ is not amenable, and in the past decade, $ B\ell^p$ was found tobe 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 $B\ell^p$ forall $p \in 1,\infty$.

Author: Volker Runde


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