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Let X be a compact Hausdorff space and let τ be a topological involution on X. In 1988, Kulkarni and Arundhathi studied Choquet and Shilov boundaries for real uniform function algebras on (X,τ). Then in 2000, Kulkarni and Limaye studied the concept of boundaries and Choquet sets for uniformly closed real subspaces and subalgebras of C(X,τ) or C(X). In 1971, Dales obtained some properties of peak sets and p-sets for complex Banach function algebras on X. Later in 1990, Arundhathi presented some results on peak sets for real uniform function algebras on (X,τ). In this paper, while we present a brief account of the work of others, we extend some of their results, either to real subspaces of C(X,τ) or to real Banach function algebras on (X,τ).

Choquet and Shilov Boundaries, Peak Sets, and Peak Points for Real Banach Function Algebras