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

We endow the Banach-Lie structure group $Str\left(V\right)$ of an infinite dimensional JB-algebra 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 $G\left(\Omega \right)$, the group of transformations that preserve the positive cone Ω of the algebra V, and to $Aut\left(V\right)$, the group of Jordan automorphisms of the algebra. We present the cone Ω as a homogeneous space for the action of $G\left(\Omega \right)$, therefore inducing a quotient Finsler metric and distance. With the techniques introduced, we prove the minimality 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 same distance there, which helps us prove the minimality of certain paths in $G\left(\Omega \right)$ for its left-invariant Finsler metric.