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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Characterizations of amenable Banach algebras


Author: Anthony To-Ming Lau
Journal: Proc. Amer. Math. Soc. 70 (1978), 156-160
MSC: Primary 46L05; Secondary 46H05
DOI: https://doi.org/10.1090/S0002-9939-1978-0492065-4
MathSciNet review: 492065
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Abstract: In this paper we show that a Banach algebra A is amenable if and only if A has any one of the following properties: (i) whenever X is a Banach A-bimodule and Y is an A-submodule of X, then for each $ f \in {Y^ \ast }$ such that $ a \cdot f = f \cdot a,a \in A$, there exists $ \tilde f \in {X^ \ast }$ which extends f and $ a \cdot \tilde f = \tilde f \cdot a$ for all $ a \in A$; (ii) whenever X is a Banach A-bimodule, there exists a bounded projection P from $ {X^ \ast }$ onto { $ f \in {X^ \ast };a \cdot f = f \cdot a$ for all $ a \in A$} such that $ T \cdot P = P \cdot T$ for any $ \mathrm{weak}^*$ continuous bounded linear operator T from $ {X^ \ast }$ into $ {X^\ast}$ commuting with the action of A on $ {X^\ast}$. The class of ultraweakly amenable von Neumann algebras with separable predual can be similarly characterized.


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DOI: https://doi.org/10.1090/S0002-9939-1978-0492065-4
Keywords: Amenable Banach algebra, amenable group, injective von Neumann algebra, Hahn-Banach extension
Article copyright: © Copyright 1978 American Mathematical Society