Loos symmetric cones

In this paper we consider Loos symmetric spaces on an open cone Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} in the Banach space setting and develop the foundations of a geometric theory based on the (modified) Loos axioms for such cones. In particular we establish exponential and log functions that exhibit many desirable features reminiscent of those of the exponential function from the space of self-adjoint elements to the cone of positive elements in a unital C∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^*$$\end{document}-algebra. We also show that the Thompson metric arises as the distance function for a natural Finsler structure on Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} and its minimal geodesics agree with the geodesics of the spray arising from the Loos structure. We close by showing that some familiar operator inequalities can be derived in this very general setting using the differential and metric geometry of the cone.