C⁎-convexity of norm unit balls

We characterize the norms on B(H)B(H) whose unit balls are C⁎C⁎-convex. We call such norms M-norms and investigate their dual norms, named L-norms. We show that the class of L-norms greater than a given norm enjoys a minimum element and 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 and L  -norms on B(H)B(H).