\(\mathcal {B}(\ell ^p)\) 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.


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
Full Text: DOI arXiv


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