Siegel domains over Finsler symmetric cones

TL;DR

This paper characterizes when tube domains over proper open cones in Banach spaces are biholomorphic to bounded symmetric domains, linking this to Finsler symmetric cones and JB-algebras.

Contribution

It establishes a precise equivalence between biholomorphic symmetry of tube domains and the cone being a Finsler symmetric cone associated with JB-algebras.

Findings

Tube domains over certain cones are biholomorphic to bounded symmetric domains.

Finsler symmetric cones correspond to the interior of squares in JB-algebras.

Characterization of symmetric cones via Banach space and algebraic structures.

Abstract

Let $\Omega$ be a proper open cone in a real Banach space $V$. We show that the tube domain $V \oplus i\Omega$ over $\Omega$ is biholomorphic to a bounded symmetric domain if and only if $\Omega$ is a normal linearly homogeneous Finsler symmetric cone, which is equivalent to the condition that $V$ is a unital JB-algebra in an equivalent norm and $\Omega$ is the interior of $\{v^2: v\in V\}$.