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 the diameter two, where $\tau$ 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 $\tau$ 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. As a result, the similar results also hold for an infinite dimensional uniform algebra.