Connections and Finsler geometry of the structure group of a JB-algebra

Abstract

We endow the Banach-Lie structure group Str(V) of an infinite dimensional JB-algebra V with a left-invariant connection and Finsler metric, and wecompute all the quantities of its connection. We show how this connection reduces toG(Ω), the group of transformations that preserve the positive cone Ω of the algebraV , and to Aut(V ), the group of Jordan automorphisms of the algebra. We presentthe cone Ω as a homogeneous space for the action of G(Ω), therefore inducing aquotient Finsler metric and distance. With the techniques introduced, we prove theminimality of the one-parameter groups in Ω for any symmetric gauge norm in V .We establish that the two presentations of the Finsler metric in Ω give the samedistance there, which helps us prove the minimality of certain paths in G(Ω) for itsleft-invariant Finsler metric.