Choquet and Shilov boundaries, peak sets, and peak points for real Banach function algebras
Let \(X\) be a compact Hausdorff space and \(\tau\) be a topological involution on \(X\), meaning that \(\tau \circ \tau\) is the identity map on \(X\). A class of real Banach algebras of continuous complex-valued functions \(C(X,\tau)\) was defined by \textit{S. H. Kulkarni} and \textit{B. V. Limaye} [Can. J. Math. 33, 181--200 (1981; Zbl 0469.46042)]. Namely, \(C(X,\tau)\) is the set \(\{f\in C(X): f(\tau(x))= \overline{f(x)}\}\), where \(C(X)\) denotes the usual Banach algebra of all continuous complex-valued functions on \(X\) equipped with the sup-norm. Then \(C(X,\tau)\) is uniformly closed, separates points, and contains the real constants. Any subalgebra \(A\) of \(C(X,\tau)\) with these same three properties is called a real function algebra on \((X,\tau)\). The concepts of Choquet and Shilov boundaries, peak points and \(p\)-points for real uniform function algebras were studied, respectively, by \textit{S. H. Kulkarni} and \textit{S. Arundhathi} [Can. J. Math. 40, No. 5, 1084--1104 (1988; Zbl 0652.46049)] and by \textit{S. Arundhathi} [Indian J. Pure Appl. Math. 21, No. 2, 155--162 (1990; Zbl 0701.46041)]. Later, the notions of Choquet sets and boundaries for real subspaces of \(C(X)\) and \(C(X, \tau)\) were introduced by \textit{S. H. Kulkarni} and \textit{B. V. Limaye} [Math. Jap. 51, No. 2, 199--212 (2000; Zbl 0989.46031)]. They obtained interesting results on the Choquet and Shilov boundaries for uniformly closed real subalgebras of \(C(X)\) and \(C(X,\tau)\), as well as the relations between the Choquet boundary of a real uniform function algebra and the Choquet boundary of its complexification. On the other hand, \textit{H. G. Dales} [Proc. Lond. Math. Soc., III. Ser. 22, 121--136 (1971; Zbl 0211.15902); Bull. Lond. Math. Soc. 3, 204--208 (1971; Zbl 0216.41302)] obtained results on the boundaries and peak sets for complex Banach function algebras. In this paper, a great amount of those known results are presented and some of these results are extended to real subspaces of \(C(X,\tau)\) or to real Banach function algebras on \((X,\tau)\).
scientific article; zbMATH DE number 6167902
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