\(C^\ast\)-convexity and \(C^\ast\)-faces in \(\ast\)-rings (Q2892203)

In this paper, the authors extend the notion of \(C^\ast\)-convexity, \(C^\ast\)-extreme point and \(C^\ast\)-face as a form of non-commutative convexity to \(^\ast\)-rings and study some of their properties. The authors try to investigate the role of algebraic structure in the geometric structure of \(C^\ast\)-convexity through the \(^\ast\)-rings. They introduce the notion of \(C^\ast\)-convex map on \(C^\ast\)-convex subsets of \(^\ast\)-ring and identify optimal points of some unital \(^\ast\)-homomorphisms on some \(C^\ast\)-convex sets. The authors show that Krein-Milman type theorems can be extended to \(C^\ast\)-convex sets of \(^\ast\)-rings. The results are interesting and contribute to generalized convexity spaces in non-commutative algebraic structures.