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$\mathcal {B}(\ell ^p)$ is never amenable


Author: Volker Runde
Journal: J. Amer. Math. Soc. 23 (2010), 1175-1185
MSC (2010): Primary 47L10; Secondary 46B07, 46B45, 46H20
DOI: https://doi.org/10.1090/S0894-0347-10-00668-5
Published electronically: March 26, 2010
MathSciNet review: 2669711
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Abstract: We show that if $E$ is a Banach space with a basis satisfying a certain condition, then the Banach algebra $\ell ^\infty ({\mathcal K}(\ell ^2 \oplus E))$ is not amenable; in particular, this is true for $E = \ell ^p$ with $p \in (1,\infty )$. As a consequence, $\ell ^\infty ({\mathcal K}(E))$ is not amenable for any infinite-dimensional ${\mathcal L}^p$-space. This, in turn, entails the non-amenability of ${\mathcal B}(\ell ^p(E))$ for any ${\mathcal L}^p$-space $E$, so that, in particular, ${\mathcal B}(\ell ^p)$ and ${\mathcal B}(L^p[0,1])$ are not amenable.


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Additional Information

Volker Runde
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
Email: vrunde@ualberta.ca

Keywords: Amenability, Kazhdan’s property $(T)$, ${\mathcal L}^p$-spaces
Received by editor(s): July 4, 2009
Received by editor(s) in revised form: December 5, 2009, December 7, 2009, and December 8, 2009
Published electronically: March 26, 2010
Additional Notes: The author’s research was supported by NSERC
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.