Ordinary algebras of complex-valued functions

Abstract

A classical theorem due to Gleason, Kahane, and Zelazko states that every complex unital Banach algebra is ordinary, meaning that any linear functional mapping elements into their spectrum is a character. While Roitman and Sternfeld extended this property to algebras with bounded spectra-showing in particular that the algebra C(X) of continuous functions is ordinary when X is pseudocompact-the general case of C(X) for arbitrary topological spaces X appears to be open, as we could not locate any reference in this direction. In this paper, we show that the pseudocompactness condition is unnecessary, that is, C(X) is ordinary for every topological space X. This fact is obtained as a consequence of a more general result characterizing positive characters on a purely algebraic generalization of unital C*-algebras, which we introduce under the name of AC*-algebras. As applications, we obtain two necessary and sufficient conditions for a Tychonoff space X to be realcompact in terms of 1-evaluating linear functionals on C(X) and their continuity with respect to the compact-open topology. Finally, we establish a Banach--Stone type theorem showing that two realcompact spaces X and Y are homeomorphic if and only if C(X) and C(Y) are linearly isomorphic via a map that preserves invertibility.