Generalized Strong Boundary Points and Boundaries of Families of Continuous Functions

Abstract

If A is a family of continuous functions on a locally compact Hausdorff space X, a boundary for A is a subset B ⊂ X such that every f ∈ A attains its maximum modulus on B. In this work we generalize the concept of strong boundary points for families of functions and show that the collection of these generalized strong boundary points is always a boundary for A. We give conditions under which all boundaries for A consist of generalized strong boundary points and under which these points coincide with classical strong boundary points. When A has sufficient algebraic structure it is proven that this construction provides a unique boundary for A consisting of boundary points, and we conclude by demonstrating how this approach provides an alternate technique for proving the existence of the Choquet and Shilov boundaries in certain function algebras.Mathematics Subject Classification (2010). Primary 46J10, 46J20; Secondary 46H40.

Maps between Banach function algebras satisfying certain norm conditions

“…It should be noted that for a locally compact Hausdorff space X and A ⊆ C 0 (X ) such intersections were considered in [13]. Indeed, in [13] a non-empty subset E of X is called an -set for A if there exists a subset F of A such that E = ∈F M .…”

“…Indeed, in [13] a non-empty subset E of X is called an -set for A if there exists a subset F of A such that E = ∈F M . In particular, the subsets I of X considered above are all -sets.…”

“…By [13, Lemma 3.2, Corollary 3.3] every -set E for a subset A of C 0 (X ) contains a minimal -set of A and the sets containing at least one point of each minimal -set (for any choice) are boundaries for A. Moreover, by [13] in the case where A is a subalgebra of C 0 (X ), an -set E for A is minimal if and only if it is a p-set, that is E = α M α for a family of peaking functions { α } of A. Thus, in this case, for each ∈ E, α ∈ F (A) and consequently I ⊆ E, and it follows from the minimality of E that E = I .…”