Geometry of unitaries in a finite algebra: variation formulas and convexity

Abstract

Given a C∗-algebra A with trace τ, we compute the first and second variation formulas for the p-energy functional Fp of the unitary group UA of A, for p = 2n an even integer, namely: Fp(γ) = Z b a τ([ ˙γ∗γ˙ ] n)dt, where γ(t) ∈ UA 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 xp = τ([x∗x] n)1/p, u0, u1, u2 ∈ UA with ui − uj < 1 2 p2 − √2 and δ(t) is a geodesic of UA joining δ(0) = u0 and δ(1) = u1, then the mapping f(t) = dp(u2, δ(t))p, t ∈ [0, 1] 15 is convex.