Large C*-polyhedrons in C*-algebras

A classical polyhedron in a Banach space is a collection of points with a distinctive geometric separation property: each point in the set can be separated from the others by a closed convex set. This concept reflects the interplay between convexity and the geometry of Banach spaces. In this article, we introduce and study a noncommutative analogue of this notion, based on the concept of $C^{⁎}$-convexity, a generalization of classical convexity within the setting of $C^{⁎}$-algebras. We define the notion of a $C^{⁎}$-polyhedron as a family of elements in a $C^{⁎}$-algebra that satisfies a similar separation property with respect to closed $C^{⁎}$-convex sets. Our main goal is to investigate the maximal possible cardinality of $C^{⁎}$-polyhedrons in various $C^{⁎}$-algebras, with particular attention to classical examples from the theory of operator algebras, such as the algebras of compact and bounded operators on a Hilbert space.