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Unit Elements in the Double Dual of a Subalgebra of the Fourier Algebra A(G)
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- Journal:
- Canadian Journal of Mathematics / Volume 61 / Issue 2 / 01 April 2009
- Published online by Cambridge University Press:
- 20 November 2018, pp. 382-394
- Print publication:
- 01 April 2009
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- Article
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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 
${{\mathcal{A}}^{**}}$.
