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

We endow the Banach-Lie structure group Str(V) of an infinite dimensional JBalgebra V with a left-invariant connection and Finsler metric, and we compute all the quantities of its connection. We show how this connection reduces to G(Q), the group of transformations that preserve the positive cone Q of the algebra V, and to Aut(V), the group of Jordan automorphisms of the algebra. We present the cone Q as a homogeneous space for the action of G(Q), therefore inducing a quotient Finsler metric and distance. With the techniques introduced, we prove the minimality of the one-parameter groups in Q for any symmetric gauge norm in V. We establish that the two presentations of the Finsler metric in Q give the same distance there, which helps us prove the minimality of certain paths in G(Q) for its left-invariant Finsler metric. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.