Closed Convex Hulls and Polyhedral Approximation in C^* - Algebras and Banach Spaces with Nontrivial Twisted Sums

Abstract

We study closed convex hulls of unitary orbits in various C^*-algebras. For unital C^*-algebras with real rank zero and a faithful tracial state determining equivalence of projections, a notion of majorization which describes the closed convex hulls of unitary orbits for self-adjoint operators are considered. Other notions of majorization are examined in these C^*-algebras. We show that norms on certain Banach spaces can be approximated uniformly, and with arbitrary precision, on bounded subsets by C^∞ smooth norms and polyhedral norms. We employ the pinching theorem, ensuring that some operators admit any sequence of contractions as an operator diagonal. Nontrivial twisted are shown.