This is an announcement for the paper "(Non-)amenability of B(E)" by Volker Runde.

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 $p \in [1,\infty]$.

Archive classification: math.FA math.HO

Mathematics Subject Classification: Primary 47L10; Secondary 46B07, 46B45, 46H20

Remarks: 16 pages; a survey article

The source file(s), BE.tex: 42631 bytes

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/0909.2628

or

http://arXiv.org/abs/0909.2628

or by email in unzipped form by transmitting an empty message with subject line

uget 0909.2628

or in gzipped form by using subject line

get 0909.2628

to: math@arXiv.org.