Factorization and existence of the unit in commutative Banach algebra (Q1568649)

A commutative Banach algebra \(A\) is said to satisfy the property of factorization of bounded sequences if for all bounded sequence \((x_n)\) in \(A\) there are \(a\in A\) and a bounded sequence \((y_n)\) in \(A\) with \(x_n= ay_n\) for all \(n\). The following two results are proved: (1) Let \(A\) be a commutative Banach algebra and let \(a\in A\) satisfy that \(aA\) is dense in \(A\). For all \(b\in A\) there is a bounded sequence \((b_n)\) in \(A\) such that \((ab_n)\) converges to \(b\) if and only if \(A\) has a unit. This theorem permits the authors to obtain extensions of results of \textit{P. G. Dixon} [Math. Proc. Cambridge Phil. Soc. 107, No. 3, 557-571 (1990; Zbl 0723.46034)] and \textit{V. Runde} [Arch. Math. 58, No. 2, 183-189 (1992; Zbl 0759.46045)]. (2) If a commutative Banach algebra \(A\) verifying the property of factorization of bounded sequences has an element \(a\in A\) such that \(aA\) is dense in \(A\), then \(A\) has a unit.