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

Source abstract

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