Diameter two properties in some vector-valued function spaces

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, τ )) over an infinite dimensional uniform algebra has the diameter two, where τ is a locally convex Hausdorff topology on a Banach space X compatible to a dual pair. Under the assumption on X being uniformly convex with norm topology τ and the additional condition that A⊗X ⊂ A(K,X), it is shown that Daugavet points and ∆-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. As a result, the similar results also hold for an infinite dimensional uniform algebra.