Functional calculus for Banach function algebras and Banach function spaces of continuous functions vanishing at infinity (Q1375928)

Let \(X\) be a locally compact Hausdorff space and \(C_0(X)\) the Banach space of all continuous real- or complex-valued functions defined on \(X\) which vanish at infinity, endowed with the sup norm. A Banach function algebra on \(X\) is a subalgebra \(B\) of \(C_0(X)\) which is a Banach algebra in some norm. The author gives several conditions so that \(B= C_0(X)\). For example, it is shown that: (1) If \(b^{1/p}\in B\) for each \(b\in B\), where \(p\) is an odd natural number larger than 1, then \(B= C_0(X)\). (2) If \(| b|^s\in B\) for each \(b\in B\), where \(0< s< 1\), then \(B= C_0(X)\). These two results are also true if \(B\) is a Banach function space of continuous real-valued functions vanishing at infinity on \(X\).