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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
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

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.

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