Non-amenability of ${\mathcal {B}}(E)$ has been surprisingly difficult to prove for the classical Banach spaces, but is now known for E = ℓp and E = Lp for all 1 ⩽ p < ∞. However, the arguments are rather indirect: the proof for L1 goes via non-amenability of $\ell ^\infty ({\mathcal {K}}(\ell _1))$ and a transference principle developed by Daws and Runde (Studia Math., 2010).In this note, we provide a short proof that ${\mathcal {B}}(L_1)$ and some of its subalgebras are non-amenable, which completely bypasses all of this machinery. Our approach is based on classical properties of the ideal of representable operators on L1, and shows that ${\mathcal {B}}(L_1)$ is not even approximately amenable.

中文翻译：

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ℬ(L1) 不适合的简短证明

不顺从${\mathcal {B}}(E)$证明经典的 Banach 空间出奇地困难，但现在已知乙= ℓp和乙=大号p对于所有 1 ⩽p< ∞。然而，这些论点是相当间接的：大号1通过非顺从性$\ell ^\infty ({\mathcal {K}}(\ell _1))$以及由 Daws 和 Runde 开发的移情原理（Studia Math., 2010）。在本笔记中，我们提供了一个简短的证明：${\mathcal {B}}(L_1)$它的一些子代数是不适合的，它完全绕过了所有这些机制。我们的方法基于可表示运算符理想的经典属性大号1，并表明${\mathcal {B}}(L_1)$甚至几乎不适合。