The Wedderburn decomposability of some commutative Banach algebras (Q1198572)

A Banach algebra \(A\) is said to have Wedderburn decomposition if \(A=B\oplus\text{rad} A\), the direct sum of a subalgebra \(B\) of \(A\) and the radical. If \(B\) is closed, then \(A\) is said to have strong Wedderburn decomposition. In this paper, the authors are concerned with the situation in which the Fourier algebra \(A(G)\) of a locally compact abelian group \(G\) fails to have Wedderburn decomposition. Specifically, the quotient algebra \(A(G)/\overline{J(E)}\) does not have Wedderburn decomposition in which \(E\) is a closed subset of nonsynthesis in \(G\) and \(\overline{J(E)}\) is the closure of the smallest ideal of \(A(G)\) with its hull equal to \(E\). Certain Beurling algebras also have the same property.