Generalized Strong Boundary Points and Boundaries of Families of Continuous Functions.

If $${\mathcal{A}}$$ is a family of continuous functions on a locally compact Hausdorff space X, a boundary for $${\mathcal{A}}$$ is a subset $${B \subset X}$$ such that every $${f \in \mathcal{A}}$$ 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 $${\mathcal{A}}$$ . We give conditions under which all boundaries for $${\mathcal{A}}$$ consist of generalized strong boundary points and under which these points coincide with classical strong boundary points. When $${\mathcal{A}}$$ has sufficient algebraic structure it is proven that this construction provides a unique boundary for $${\mathcal{A}}$$ 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. [ABSTRACT FROM AUTHOR]