Factorization in some ideals of Lau algebras with applications to semigroup algebras (Q1590251)

Let \(A\) be a complex Banach algebra which, as a Banach space, is the predual of a von Neumann algebra \(M\) and has the property that the identity element \(u\) of \(M= A^*\) is a multiplicative linear functional on \(A\). Such Banach algebras have first been studied by \textit{A. T.-M. Lau} [Fundam. Math. 118, 161-175 (1983; Zbl 0545.46051)] and are therefore often referred to as Lau algebras. Examples of Lau algebras are provided by the measure algebra and the \(L^1\)-algebra, \(M(G)\) and \(L^1(G)\), of a locally compact group \(G\) and by the Fourier algebra of \(G\). Let \(A\) be a Lau algebra and \(I_0(G)\) the kernel of the homomorphism \(u\) of \(A\). For any closed ideal \(J\) of \(A\), let \(I_0(J)= J\cap I_0(A)\) and let \(J^2\) denote the set of all elements \(\sum^n_{k=1} x_k y_k,\;x_k,y_k\in J\), \(n\in\mathbb{N}\). Extending a result of \textit{G. Willis} [Proc. Am. Math. Soc. 86, 599-601 (1982; Zbl 0512.43003)] for \(M(G)\) and \(L^1(G)\), the author shows that if \(A\) is a Lau algebra with bounded approximate identity then \(I_0(A)^2= I_0(A)\), and if, in addition, \(J\) is a closed ideal of \(A\) with bounded approximate identity then \(I_0(J)^2= I_0(J)\). The proof exploits Cohen's factorization theorem and Willis' idea. An application concerns factorization of codimension one ideals in \(M_a(S)\), the analogue of \(L^1(G)\), where \(S\) is a so-called foundation semigroup with identity.