IAM   02674
INSTITUTO ARGENTINO DE MATEMATICA ALBERTO CALDERON
Unidad Ejecutora - UE
artículos
Título:
Geometry of unitaries in a finite algebra: variation formulas
Autor/es:
ESTEBAN ANDRUCHOW, L¨¢ZARO RECHT
Revista:
INTERNATIONAL JOURNAL OF MATHEMATICS
Referencias:
Año: 2008 vol. 19 p. 1223 - 1223
ISSN:
0129-167X
Resumen:
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.