2 results
A short proof that ℬ(L1) is not amenable
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 151 / Issue 6 / December 2021
- Published online by Cambridge University Press:
- 06 November 2020, pp. 1758-1767
- Print publication:
- December 2021
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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.
The Operator Amenability of Uniform Algebras
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- Journal:
- Canadian Mathematical Bulletin / Volume 46 / Issue 4 / 01 December 2003
- Published online by Cambridge University Press:
- 20 November 2018, pp. 632-634
- Print publication:
- 01 December 2003
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We prove a quantized version of a theorem by M. V. Sheĭnberg: A uniform algebra equipped with its canonical, i.e., minimal, operator space structure is operator amenable if and only if it is a commutative
${{C}^{*}}$-algebra.
