Maximal ideal spaces and 𝐴-convexity

Abstract

notes the Banach algebra of all complex valued continuous functions on X with the supremum norm. A subalgebra A contained in C(X) is called a function algebra on X if it satisfies the following conditions. (i) A separates points on X. (ii) The constant functions are in A, i.e., 1 belongs to A. (iii) A is closed in C(X). For a function algebra A the space of maximal ideals is denoted by MA and the Silov boundary by SA. Both SA and MA carry natural compact Hausdorff topologies and SA and X can be imbedded in MA so that, identifying SA and X with their images in MA, we have SACXCMA. Moreover A, via restriction, is a function algebra on SA and, via the Gelfand representation, extends to a function algebra on MA. SA and MA are respectively the smallest and largest compact Hausdorff spaces on which A can be realized as a function algebra.