MAPS PRESERVING SCHATTEN p-NORMS OF CONVEX COMBINATIONS

Abstract. In this paper, we study maps φ of positive operators of Schatten p-classes (1 < p < +∞), which preserve the p-norms of convex combinations, that is, ‖tρ+ (1 − t)σ‖p = ‖tφ(ρ) + (1 − t)φ(σ)‖p, ∀ρ, σ ∈ S+p (H)1, t ∈ [0, 1]. They are exactly those carrying the form φ(ρ) = UρU ∗ for a unitary or anti-unitary U. In the case p = 2, we have the same conclusion whenever it just holds ‖ρ+ σ‖2 = ‖φ(ρ) + φ(σ)‖2 for all positive Hilbert-Schmidt class operators ρ, σ of norm 1. Some examples are demon-strated