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 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 $G(\mathrm{\Omega })$, the group of transformations that preserve the positive cone $\mathrm{\Omega }$ of the algebra $V$, and to $Aut(V)$, the group of Jordan automorphisms of the algebra. We present the cone $\mathrm{\Omega }$ as an homogeneous space for the action of $G(\mathrm{\Omega })$, therefore inducing a quotient Finsler metric and distance. With the techniques introduced, we prove the minimality of the one-parameter groups in $\mathrm{\Omega }$ for any symmetric gauge norm in $V$. We establish that the two presentations of the Finsler metric in $\mathrm{\Omega }$ give the same distance there, which helps us prove the minimality of certain paths in $G(\mathrm{\Omega })$ for its left-invariant Finsler metric.