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

Published:2009-04-01
Printed: Apr 2009
• Tianxuan Miao
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## Abstract

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 \"Ulger, showing that for the Fourier algebra $A(G)$ of a locally compact group $G$, an element $\phi \in A(G)^{**}$ is in $A(G)$ if and only if $A(G) \phi \subseteq A(G)$ and $E \phi = \phi$ for all right identities $E$ of $A(G)^{**}$. We also prove some results about the topological centers of ${\mathcal B}^{**}$ and ${\mathcal A}^{**}$.
 Keywords: Locally compact groups, amenable groups, Fourier algebra, identity, Arens product, topological center
 MSC Classifications: 43A07 - Means on groups, semigroups, etc.; amenable groups

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