Abstract: Many problems in functional analysis are best addressed via extreme points. Extreme points, particularly in the guise of the Choquet or Shilov boundaries, play a substantial role in the theory of uniform algebras. This theory and its many applications are developed in this chapter. Topics covered include the Choquet boundary and boundary representations, the injective envelope, the C*-envelope, the multiplier algebra of an operator space, and multipliers and ‘characterization theorems’, and multipliers and duality.

Keywords: extreme points, Choquet boundary, injective envelope, C*-envelope, multiplier algebra, operator space, duality,