Geodesic
ℬ(ℓ^{𝓅}) is never amenable
Runde, Volker1
1 Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
Journal of the American Mathematical Society, Tome 23 (2010) no. 4, pp. 1175-1185

Voir la notice de l'article provenant de la source American Mathematical Society

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.
DOI : 10.1090/S0894-0347-10-00668-5

Runde, Volker 1

1 Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1 
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Runde, Volker. ℬ(ℓ^{𝓅}) is never amenable. Journal of the American Mathematical Society, Tome 23 (2010) no. 4, pp. 1175-1185. doi: 10.1090/S0894-0347-10-00668-5

[1] Bekka, Bachir, De La Harpe, Pierre, Valette, Alain Kazhdan’s property (T) 2008

[2] Daws, Matthew, Runde, Volker Can ℬ(𝓁^{𝓅}) ever be amenable? Studia Math. 2008 151 174

[3] Diestel, Joe, Jarchow, Hans, Tonge, Andrew Absolutely summing operators 1995

[4] Gordon, Yehoram On 𝑝-absolutely summing constants of Banach spaces Israel J. Math. 1969 151 163

[5] Grã¸Nbã¦K, N., Johnson, B. E., Willis, G. A. Amenability of Banach algebras of compact operators Israel J. Math. 1994 289 324

[6] Helemskii, A. Ya. The homology of Banach and topological algebras 1989

[7] Johnson, Barry Edward Cohomology in Banach algebras 1972

[8] Johnson, B. E. Approximate diagonals and cohomology of certain annihilator Banach algebras Amer. J. Math. 1972 685 698

[9] Lindenstrauss, J., Peå‚Czyå„Ski, A. Absolutely summing operators in 𝐿_{𝑝}-spaces and their applications Studia Math. 1968 275 326

[10] Lindenstrauss, Joram, Tzafriri, Lior Classical Banach spaces. I 1977

[11] Ozawa, Narutaka A note on non-amenability of ℬ(𝓁_{𝓅}) for 𝓅 Internat. J. Math. 2004 557 565

[12] Pisier, G. On Read’s proof that 𝐵(𝑙₁) is not amenable 2004 269 275

[13] Read, C. J. Commutative, radical amenable Banach algebras Studia Math. 2000 199 212

[14] Read, C. J. Relative amenability and the non-amenability of 𝐵(𝑙¹) J. Aust. Math. Soc. 2006 317 333

[15] Runde, Volker The structure of contractible and amenable Banach algebras 1998 415 430

[16] Runde, Volker Lectures on amenability 2002

[17] Tomczak-Jaegermann, Nicole Banach-Mazur distances and finite-dimensional operator ideals 1989

[18] Wassermann, Simon On tensor products of certain group 𝐶*-algebras J. Functional Analysis 1976 239 254

[19] Wojtaszczyk, P. Banach spaces for analysts 1991

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