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Summer 2016 Amenability properties of the central Fourier algebra of a compact group
Mahmood Alaghmandan, Nico Spronk
Illinois J. Math. 60(2): 505-527 (Summer 2016). DOI: 10.1215/ijm/1499760019

Abstract

We let the central Fourier algebra, $\operatorname{ZA}(G)$, be the subalgebra of functions $u$ in the Fourier algebra $\mathrm{A}(G)$ of a compact group, for which $u(xyx^{-1})=u(y)$ for all $x$, $y$ in $G$. We show that this algebra admits bounded point derivations whenever $G$ contains a non-Abelian closed connected subgroup. Conversely when $G$ is virtually Abelian, then $\operatorname{ZA}(G)$ is amenable. Furthermore, for virtually Abelian $G$, we establish which closed ideals admit bounded approximate identities. We also show that $\operatorname{ZA}(G)$ is weakly amenable, in fact hyper-Tauberian, exactly when $G$ admits no non-Abelian connected subgroup. We also study the amenability constant of $\operatorname{ZA}(G)$ for finite $G$ and exhibit totally disconnected groups $G$ for which $\operatorname{ZA}(G)$ is non-amenable. In passing, we establish some properties related to spectral synthesis of subsets of the spectrum of $\operatorname{ZA}(G)$.

Citation

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Mahmood Alaghmandan. Nico Spronk. "Amenability properties of the central Fourier algebra of a compact group." Illinois J. Math. 60 (2) 505 - 527, Summer 2016. https://doi.org/10.1215/ijm/1499760019

Information

Received: 2 September 2015; Revised: 13 March 2017; Published: Summer 2016
First available in Project Euclid: 11 July 2017

zbMATH: 1369.43002
MathSciNet: MR3680545
Digital Object Identifier: 10.1215/ijm/1499760019

Subjects:
Primary: 43A30, 43A77
Secondary: 22D45, 43A45, 46H25, 46J40

Rights: Copyright © 2016 University of Illinois at Urbana-Champaign

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Vol.60 • No. 2 • Summer 2016
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