On Šilov boundary for function spaces
This paper is an attempt to generalize classical boundary theory for subspaces of the unital commutative \(C^*\)-algebra \(C(X)\) of continuous complex-valued functions on the compact Hausdorff space \(X\) that contain the unit in \(C(X)\) to those that do not. It owes much to the classic monograph of \textit{R. R. Phelps} entitled `Lectures on Choquet's theorem' [Princeton etc.: Van Nostrand (1966; Zbl 0135.36203); Berlin: Springer (2001; Zbl 0997.46005)] and to his paper [Bull. Am. Math. Soc. 83, 299--312 (1977; Zbl 0382.46005)]. In particular, it is shown that, for closed subspaces \(A\) of \(C(X)\) that separate points in \(X\) but do not contain the unit in \(C(X)\), the closure of the Choquet boundary \(\partial A\) is a `bundle' that coincides with the Šilov boundary `bundle'.
scientific article; zbMATH DE number 6474009
scientific article