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arXiv · 2609.02680

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

Neha Hotwani·T. S. S. R. K. Rao
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Abstract

Let \(\Omega\) 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 \(C(\Omega, \cA)\), 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 \(C(\Omega, \cA)_1\) is a unitary. Additionally, we identify a classes of \(C^*\)-algebras where the concepts of \(C^*\)-extreme and pointwise \(C^*\)-extreme points in \(C(\Omega, \cA)_1\) coincide. We show this holds if a von Neumann algebra has a separable predual with the Radon-Nikod\'ym property.

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Neha Hotwani, T. S. S. R. K. Rao. 2026-09-02. Vector valued continuous function spaces as $C^\ast$-algebras. https://arxiv.org/abs/2609.02680

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