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Weak amenability of commutative Beurling algebras


Author: Yong Zhang
Journal: Proc. Amer. Math. Soc. 142 (2014), 1649-1661
MSC (2010): Primary 46H20, 43A20; Secondary 43A10
DOI: https://doi.org/10.1090/S0002-9939-2014-11955-1
Published electronically: February 13, 2014
MathSciNet review: 3168471
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Abstract: For a locally compact Abelian group $ G$ and a continuous weight function $ \omega $ on $ G$ we show that the Beurling algebra $ L^1(G, \omega )$ is weakly amenable if and only if there is no nontrivial continuous group homomorphism $ \phi $: $ G\to \mathbb{C}$ such that $ \sup _{t\in G}\frac {\vert\phi (t)\vert}{\omega (t)\omega (t^{-1})} < \infty $. Let $ \widehat \omega (t) = \limsup _{s\to \infty }\omega (ts)/\omega (s)$ ($ t\in G$). Then $ L^1(G, \omega )$ is $ 2$-weakly amenable if there is a constant $ m> 0$ such that $ \liminf _{n\to \infty }\frac {\omega (t^n)\widehat \omega (t^{-n})}{n} \leq m$ for all $ t\in G$.


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Additional Information

Yong Zhang
Affiliation: Department of Mathematics, University of Manitoba, Winnipeg R3T 2N2, Canada
Email: zhangy@cc.umanitoba.ca

DOI: https://doi.org/10.1090/S0002-9939-2014-11955-1
Keywords: Derivation, weak amenability, $2$-weak amenability, weight, locally compact Abelian group
Received by editor(s): March 20, 2012
Received by editor(s) in revised form: June 7, 2012
Published electronically: February 13, 2014
Additional Notes: The author was supported by NSERC 238949-2011.
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2014 American Mathematical Society

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