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SOME CONVERGENCE THEOREMS IN FOURIER ALGEBRAS

Part of: Abstract harmonic analysis Commutative Banach algebras and commutative topological algebras

Published online by Cambridge University Press:  03 July 2017

HEYBETKULU MUSTAFAYEV Open the ORCID record for HEYBETKULU MUSTAFAYEV [Opens in a new window]
HEYBETKULU MUSTAFAYEV*
Affiliation:
Department of Mathematics, Faculty of Science, Yuzuncu Yil University, Van, Turkey email hsmustafayev@yahoo.com
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Abstract

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Let $G$ be a locally compact amenable group and $A(G)$ and $B(G)$ be the Fourier and the Fourier–Stieltjes algebras of $G,$ respectively. For a power bounded element $u$ of $B(G)$, let ${\mathcal{E}}_{u}:=\{g\in G:|u(g)|=1\}$. We prove some convergence theorems for iterates of multipliers in Fourier algebras.

(a) If $\Vert u\Vert _{B(G)}\leq 1$, then $\lim _{n\rightarrow \infty }\Vert u^{n}v\Vert _{A(G)}=\text{dist}(v,I_{{\mathcal{E}}_{u}})\text{ for }v\in A(G)$, where $I_{{\mathcal{E}}_{u}}=\{v\in A(G):v({\mathcal{E}}_{u})=\{0\}\}$.

(b) The sequence $\{u^{n}v\}_{n\in \mathbb{N}}$ converges for every $v\in A(G)$ if and only if ${\mathcal{E}}_{u}$ is clopen and $u({\mathcal{E}}_{u})=\{1\}.$

(c) If the sequence $\{u^{n}v\}_{n\in \mathbb{N}}$ converges weakly in $A(G)$ for some $v\in A(G)$, then it converges strongly.

Keywords

locally compact groupFourier algebraFourier–Stieltjes algebraconvergence

MSC classification

Primary: 43A30: Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
Secondary: 46J20: Ideals, maximal ideals, boundaries
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 96 , Issue 3 , December 2017 , pp. 487 - 495
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

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