Diameter two properties in some vector-valued function spaces

Abstract

We introduce a vector-valued version of a uniform algebra, called the vector-valued function space over a uniform algebra. The diameter two properties of the vector-valued function space over a uniform algebra on an infinite compact Hausdorff space are investigated. Every nonempty relatively weakly open subset of the unit ball of a vector-valued function space $$A(K, (X, \tau ))$$ over an infinite dimensional uniform algebra has diameter two, where $$\tau $$ is a locally convex Hausdorff topology on a Banach space X compatible to a dual pair. Under the assumption of X equipped with the norm topology being uniformly convex and the additional condition that $$A\otimes X\subset A(K, X)$$ , it is shown that Daugavet points and $$\Delta $$ -points on A(K, X) over a uniform algebra A are the same, and they are characterized by the norm-attainment at a limit point of the Shilov boundary of A. In addition, a sufficient condition for the convex diametral local diameter two property of A(K, X) is also provided. Similar results also hold for an infinite dimensional uniform algebra.