C*-Convexity of Norm Unit Balls

TL;DR

This paper characterizes norms on B(H) with C*-convex unit balls, introduces M-norms, and explores their extremal properties and constructive methods for their determination.

Contribution

It defines M-norms as norms with C*-convex unit balls, analyzes their extremal elements, and provides a constructive approach to obtain such norms.

Findings

Existence of maximum M-norm less than a given norm

Identification of minimum and maximum M-norms in certain cases

Constructive method to generate M-norms on B(H)

Abstract

We determine those norms on B(H) whose unit ball is C*-convex. We call them M-norms and show that the class of M-norms less than a given norm enjoys a maximum element. These minimum and maximum elements will be determined in some cases. Finally, we give a constructive result to obtain M-norms on B(H).