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

TL;DR

This paper explores the geometric structure of the structure group of a JB-algebra, introducing a Finsler metric and connection, and analyzing minimal paths and distances within this framework.

Contribution

It develops a Finsler geometric framework for the structure group of JB-algebras, including connections, reductions, and minimality results for paths and distances.

Findings

Established a left-invariant Finsler metric on the structure group

Proved the minimality of one-parameter groups in the positive cone

Showed the equivalence of two Finsler metrics in the cone

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