Rights statement: https://www.cambridge.org/core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics/article/short-proof-that-l1-is-not-amenable/5EB9501F273D24ABD38EDEAB8B9433AD The final, definitive version of this article has been published in the Journal, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 151 (6), pp 1758-1767 2021, © 2020 Cambridge University Press.
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Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
}
TY - JOUR
T1 - A short proof that B(L_1) is not amenable
AU - Choi, Yemon
N1 - https://www.cambridge.org/core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics/article/short-proof-that-l1-is-not-amenable/5EB9501F273D24ABD38EDEAB8B9433AD The final, definitive version of this article has been published in the Journal, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 151 (6), pp 1758-1767 2021, © 2020 Cambridge University Press.
PY - 2021/12/31
Y1 - 2021/12/31
N2 - Non-amenability of ${\mathcal B}(E)$ has been surprisingly difficult to prove for the classical Banach spaces, but is now known for $E= \ell_p$ and $E=L_p$ for all $1\leq p<\infty$. However, the arguments are rather indirect: the proof for $L_1$ 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 $L_1$, and shows that ${\mathcal B}(L_1)$ is not even approximately amenable.
AB - Non-amenability of ${\mathcal B}(E)$ has been surprisingly difficult to prove for the classical Banach spaces, but is now known for $E= \ell_p$ and $E=L_p$ for all $1\leq p<\infty$. However, the arguments are rather indirect: the proof for $L_1$ 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 $L_1$, and shows that ${\mathcal B}(L_1)$ is not even approximately amenable.
KW - Amenable Banach algebra
KW - Banach spaces
KW - operator ideals
U2 - 10.1017/prm.2020.79
DO - 10.1017/prm.2020.79
M3 - Journal article
VL - 151
SP - 1758
EP - 1767
JO - Proceedings of the Royal Society of Edinburgh: Section A Mathematics
JF - Proceedings of the Royal Society of Edinburgh: Section A Mathematics
SN - 0308-2105
IS - 6
ER -