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\(\mathcal {B}(\ell ^p)\) is never amenable. (English) Zbl 1221.47143

The paper under review finally settles the long standing open problem of non-amenability of the Banach algebra \(B(X)\) over an infinite-dimensional Banach space \(X\) belonging to a rather large class of Banach spaces including \(\ell^p\) and \(L^p[0,1]\) for \(1<p<\infty\).
Recall that a Banach algebra \(\mathcal{A}\) is amenable if every bounded derivation from \(\mathcal{A}\) to a dual \(\mathcal{A}\)-bimodule is inner. It was B. E. Johnson [“Cohomology in Banach algebras”, Mem. Am. Math. Soc. 127 (1972; Zbl 0256.18014)] who gave the above definition and made a Banach algebraic characterization of amenable locally compact groups, namely, the fact that the convolution algebra \(L^1(G)\) is amenable as a Banach algebra if and only if the locally compact group \(G\) itself is amenable. Johnson asked in his Memoir whether \(B(X)\) could ever be an amenable Banach algebra for an infinite-dimensional Banach space \(X\). The only known (non-amenable) example in this direction was \(B(\ell^2)\), obtained by heavy \(C^*\)-algebraic machinery, until C. J. Read [“Relative amenability and the non-amenability of \(B(\ell^1)\)”, J. Aust. Math. Soc. 80, No. 3, 317–333 (2006; Zbl 1111.46028)] proved that \(B(\ell^1)\) is not amenable either. The result of Read has been simplified by G. Pisier [“On Read’s proof that \(B(\ell_1)\) is not amenable”, Lecture Notes in Mathematics 1850, 269–275 (2004; Zbl 1072.46032)] and further simplified and extended to the case of \(B(\ell^p)\), \(p \in \{1,2\}\) by N. Ozawa [“A note on non-amenability of \(\mathcal{B}(\ell_p)\) for \(p=1,2\)”, Int. J. Math. 15, No. 6, 557–565 (2004; Zbl 1056.46046)].
The key idea for the proof of the non-amenability of \(B(\ell^p)\) begins with a previous result of M. Daws and the author himself [“Can \(\mathcal{B}(\ell^p)\) ever be amenable?”, Stud. Math. 188, No. 2, 151–174 (2008; Zbl 1145.47056)], saying that the amenability of \(B(\ell^p)\) implies the amenability of \(\ell^\infty(\mathcal{K}(\ell^2\oplus \ell^p))\), where \(\mathcal{K}(\cdot)\) denotes the algebra of compact operators. Then, the author makes a clever modification of N. Ozawa’s approach, which uses property (T) of \(SL(3,\mathbb{Z})\) to produce a specific sequence of unitaries, by appealing to the following result he has observed: \[ \sum_n\|Se_n\|\, \|Te^*_n\| \leq N\|S\|\, \|T\| \]
for any \(S\in B(\ell^p,\ell^2_N)\) and \(T\in B(\ell^{p'}, \ell^2_N)\), where \(e_n\) and \(e^*_n\) are canonical unit vectors of \(\ell^p\) and \(\ell^{p'}\) with \(\frac{1}{p}+\frac{1}{p'} = 1\).
The result in the paper under review is in sharp contrast to another recent result by S. A. Argyros and R. G. Haydon [“A hereditarily indecomposable \(\mathcal{L}_\infty\)-space that solves the scalar-plus-compact problem”, Acta Math. 206, No. 1, 1–54 (2011; Zbl 1223.46007)] saying that there is an infinite-dimensional Banach space \(X\) satisfying \(B(X) = \mathcal{K}(X) + \mathbb{C}I\), so that \(B(X)\) is actually an amenable Banach algebra.

MSC:

47L10 Algebras of operators on Banach spaces and other topological linear spaces
46B07 Local theory of Banach spaces
46B45 Banach sequence spaces
46H20 Structure, classification of topological algebras
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References:

[1] S. A. ARGYROS and R. G. HAYDON, A hereditarily indecomposable \( \mathcal{L}_\infty\)-space that solves the scalar-plus-compact problem. arXiv:0903.3921. · Zbl 1223.46007
[2] Bachir Bekka, Pierre de la Harpe, and Alain Valette, Kazhdan’s property (T), New Mathematical Monographs, vol. 11, Cambridge University Press, Cambridge, 2008. · Zbl 1146.22009
[3] Matthew Daws and Volker Runde, Can \Cal B(\?^{\?}) ever be amenable?, Studia Math. 188 (2008), no. 2, 151 – 174. , https://doi.org/10.4064/sm188-2-4 Matthew Daws and Volker Runde, Erratum to ”Can \Cal B(\ell ^{\?}) ever be amenable?” [MR2431000], Studia Math. 195 (2009), no. 3, 297 – 298. · doi:10.4064/sm195-3-8
[4] Joe Diestel, Hans Jarchow, and Andrew Tonge, Absolutely summing operators, Cambridge Studies in Advanced Mathematics, vol. 43, Cambridge University Press, Cambridge, 1995. · Zbl 0855.47016
[5] Yehoram Gordon, On \?-absolutely summing constants of Banach spaces, Israel J. Math. 7 (1969), 151 – 163. · Zbl 0179.17502 · doi:10.1007/BF02771662
[6] N. Grønbæk, B. E. Johnson, and G. A. Willis, Amenability of Banach algebras of compact operators, Israel J. Math. 87 (1994), no. 1-3, 289 – 324. · Zbl 0806.46058 · doi:10.1007/BF02773000
[7] A. Ya. Helemskii, The homology of Banach and topological algebras, Mathematics and its Applications (Soviet Series), vol. 41, Kluwer Academic Publishers Group, Dordrecht, 1989. Translated from the Russian by Alan West.
[8] Barry Edward Johnson, Cohomology in Banach algebras, American Mathematical Society, Providence, R.I., 1972. Memoirs of the American Mathematical Society, No. 127. · Zbl 0256.18014
[9] B. E. Johnson, Approximate diagonals and cohomology of certain annihilator Banach algebras, Amer. J. Math. 94 (1972), 685 – 698. · Zbl 0246.46040 · doi:10.2307/2373751
[10] J. Lindenstrauss and A. Pełczyński, Absolutely summing operators in \?_{\?}-spaces and their applications, Studia Math. 29 (1968), 275 – 326. · Zbl 0183.40501
[11] Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Springer-Verlag, Berlin-New York, 1977. Sequence spaces; Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 92. · Zbl 0362.46013
[12] Narutaka Ozawa, A note on non-amenability of \Cal B(\?_{\?}) for \?=1,2, Internat. J. Math. 15 (2004), no. 6, 557 – 565. · Zbl 1056.46046 · doi:10.1142/S0129167X04002430
[13] G. Pisier, On Read’s proof that \?(\?\(_{1}\)) is not amenable, Geometric aspects of functional analysis, Lecture Notes in Math., vol. 1850, Springer, Berlin, 2004, pp. 269 – 275. · Zbl 1072.46032 · doi:10.1007/978-3-540-44489-3_20
[14] C. J. Read, Commutative, radical amenable Banach algebras, Studia Math. 140 (2000), no. 3, 199 – 212. · Zbl 0972.46031
[15] C. J. Read, Relative amenability and the non-amenability of \?(\?\textonesuperior ), J. Aust. Math. Soc. 80 (2006), no. 3, 317 – 333. · Zbl 1111.46028 · doi:10.1017/S1446788700014038
[16] Volker Runde, The structure of contractible and amenable Banach algebras, Banach algebras ’97 (Blaubeuren), de Gruyter, Berlin, 1998, pp. 415 – 430. · Zbl 0927.46028
[17] Volker Runde, Lectures on amenability, Lecture Notes in Mathematics, vol. 1774, Springer-Verlag, Berlin, 2002. · Zbl 0999.46022
[18] Nicole Tomczak-Jaegermann, Banach-Mazur distances and finite-dimensional operator ideals, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 38, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989. · Zbl 0721.46004
[19] Simon Wassermann, On tensor products of certain group \?*-algebras, J. Functional Analysis 23 (1976), no. 3, 239 – 254. · Zbl 0358.46040
[20] P. Wojtaszczyk, Banach spaces for analysts, Cambridge Studies in Advanced Mathematics, vol. 25, Cambridge University Press, Cambridge, 1991. · Zbl 0724.46012
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