GEOMETRY OF UNITARIES IN A FINITE ALGEBRA: VARIATION FORMULAS AND CONVEXITY

Abstract

Given a C*-algebra [Formula: see text] with trace τ, we compute the first and second variation formulas for the p-energy functional Fp of the unitary group [Formula: see text] of [Formula: see text], for p = 2n an even integer, namely: [Formula: see text] where [Formula: see text] is a smooth curve for t ∈ [a, b]. As an application of these formulas, we prove that if dp denotes the geodesic distance of the Finsler metric induced by the p-norm [Formula: see text] with [Formula: see text] and δ(t) is a geodesic of [Formula: see text] joining δ(0) = u0 and δ(1) = u1, then the mapping [Formula: see text] is convex.