Functional Calculus for Banach Function Algebras and Banach Function Spaces of Continuous Functions Vanishing at Infinity

We show among other things that ifBis a Banach function space of continuous real-valued functions vanishing at infinity on a locally compact Hausdorff spaceX, with the property that for some odd natural numberp>1,b1/p∈Bfor allb∈B, thenB=C0(X).