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Unit Elements in the Double Dual of a Subalgebra of the Fourier Algebra A(G)

Published online by Cambridge University Press:  20 November 2018

Tianxuan Miao
Tianxuan Miao*
Affiliation:
Department of Mathematical Sciences, Lakehead University, Thunder Bay, Ontario P7E 5E1, [email protected]
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Abstract

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Let $\mathcal{A}$ be a Banach algebra with a bounded right approximate identity and let $\mathcal{B}$ be a closed ideal of $\mathcal{A}$. We study the relationship between the right identities of the double duals ${{\mathcal{B}}^{*}}^{*}$ and ${{\mathcal{A}}^{**}}$ under the Arens product. We show that every right identity of ${{\mathcal{B}}^{*}}^{*}$ can be extended to a right identity of ${{\mathcal{A}}^{**}}$ in some sense. As a consequence, we answer a question of Lau and Ülger, showing that for the Fourier algebra $A\left( G \right)$ of a locally compact group $G$, an element $\phi \in A{{\left( G \right)}^{**}}$ is in $A\left( G \right)$ if and only if $A\left( G \right)\phi \subseteq A\left( G \right)$ and $E\phi =\phi$ for all right identities $E$ of $A{{\left( G \right)}^{**}}$. We also prove some results about the topological centers of ${{\mathcal{B}}^{**}}$ and ${{\mathcal{A}}^{**}}$.

Keywords

43A07compact groupsamenable groupsFourier algebraidentityArens producttopological center.
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 61 , Issue 2 , 01 April 2009 , pp. 382 - 394
Copyright
Copyright © Canadian Mathematical Society 2009

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