Geometry of unitaries in a finite algebra: variation formulas

ESTEBAN ANDRUCHOW, L¨¢ZARO RECHT

INTERNATIONAL JOURNAL OF MATHEMATICS

Año: 2008 vol. 19 p. 1223 - 1246

0129-167X

Given a C∗-algebra A with trace ¦Ó, 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:C∗-algebra A with trace ¦Ó, 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:p-energy functional Fp of the unitary group UA of A, for p = 2n an even integer, namely: Fp(¦Ã) = Z b a ¦Ó([¦Ã¨B ∗¦Ã¨B ]n)dt,p(¦Ã) = Z b a ¦Ó([¦Ã¨B ∗¦Ã¨B ]n)dt,([¦Ã¨B ∗¦Ã¨B ]n)dt, where ¦Ã(t) ¡Ê UA is a smooth curve for t ¡Ê [a, b]. As an application of these formulas, we prove that if dp denotes the geodesic distance of the Finsler metric induced by the¦Ã(t) ¡Ê UA is a smooth curve for t ¡Ê [a, b]. As an application of these formulas, we prove that if dp denotes the geodesic distance of the Finsler metric induced by thedp denotes the geodesic distance of the Finsler metric induced by the p-norm xp = ¦Ó([x∗x]n)1/p, u0, u1, u2 ¡Ê UA with ui −uj < 1 2p2 − ¡Ì2 and ¦Ä(t) is a geodesic of UA joining ¦Ä(0) = u0 and ¦Ä(1) = u1, then the mapping-norm xp = ¦Ó([x∗x]n)1/p, u0, u1, u2 ¡Ê UA with ui −uj < 1 2p2 − ¡Ì2 and ¦Ä(t) is a geodesic of UA joining ¦Ä(0) = u0 and ¦Ä(1) = u1, then the mappingp2 − ¡Ì2 and ¦Ä(t) is a geodesic of UA joining ¦Ä(0) = u0 and ¦Ä(1) = u1, then the mappingUA joining ¦Ä(0) = u0 and ¦Ä(1) = u1, then the mapping f(t) = dp(u2, ¦Ä(t))p, t¡Ê [0, 1](t) = dp(u2, ¦Ä(t))p, t¡Ê [0, 1] 15 is convex.is convex. Keywords: Unitary element; trace; variation formulas; convexity.: Unitary element; trace; variation formulas; convexity.