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

Miao, Tianxuan

Canadian journal of mathematics, Tome 61 (2009) no. 2, pp. 382-394

Résumé

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 ${{\mathcal{A}}^{**}}$ .

Détail

DOI : 10.4153/CJM-2009-020-0

Mots-clés : 43A07, compact groups, amenable groups, Fourier algebra, identity, Arens product, topological center.

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