A short proof that ℬ(L1) is not amenable
Non-amenability of ${\mathcal {B}}(E)$ has been surprisingly difficult to prove for the classical Banach spaces, but is now known for E = ℓ p and E = L p 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.
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2010 ◽
Vol 83
(2)
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pp. 231-240
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2020 ◽
Vol 45
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pp. 863-876
1980 ◽
Vol 32
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pp. 1482-1500
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2002 ◽
Vol 45
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pp. 523-546
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2018 ◽
Vol 61
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pp. 545-555
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