×

\(C^\ast\)-extreme points and \(C^\ast\)-faces of the epigraph if \(C^\ast\)-affine maps in \(\ast\)-rings. (English) Zbl 1438.46081

Summary: In this paper, we define the notion of \(C^\ast\)-affine maps in the unital \(\ast\)-rings and we investigate the \(C^\ast\)-extreme points of the graph and epigraph of such maps. We show that for a \(C^\ast\)-convex map \(f\) on a unital \(\ast\)-ring \(\mathcal{R}\) satisfying the positive square root axiom with an additional condition, the graph of \(f\) is a \(C^\ast\)-face of the epigraph of \(f\). Moreover, we prove some results about the \(C^\ast\)-faces of \(C^\ast\)-convex sets in \(\ast\)-rings.

MSC:

46L89 Other “noncommutative” mathematics based on \(C^*\)-algebra theory
52A01 Axiomatic and generalized convexity
16W10 Rings with involution; Lie, Jordan and other nonassociative structures
PDFBibTeX XMLCite