Vector valued continuous function spaces as $C^{\ast }$-algebras

Abstract

Let \(Ω\) be a compact Hausdorff space, and let \(\cA\) be a unital \(C^*\)-algebra. In this study, we continue our examination of the comparison between \(C^*\)-extreme points and linear extremal structures of the unit ball, in the vector-valued \(C^*\)-algebra \(C(Ω, \cA)\), building upon the work initiated in \cite{HR}. We first enlarge the class of $C^{\ast }$-algebras in which a $C^{\ast }$-extreme point is an extreme point. We demonstrate that if \(\cA\) has a faithful tracial state, then any \(C^*\)-extreme point of the unit ball \(C(Ω, \cA)_1\) is a unitary. Additionally, we identify a classes of \(C^*\)-algebras where the concepts of \(C^*\)-extreme and pointwise \(C^*\)-extreme points in \(C(Ω, \cA)_1\) coincide. We show this holds if a von Neumann algebra has a separable predual with the Radon-Nikodým property.