On the connectedness of spectral sets and irreducibility of spectral cones in Euclidean Jordan algebras

TL;DR

This paper investigates the topological properties of spectral sets and cones in Euclidean Jordan algebras, establishing their connectedness and irreducibility based on eigenvalue orbit properties.

Contribution

It proves that connected permutation-invariant sets lead to connected spectral sets and that pointed spectral cones are irreducible in simple Euclidean Jordan algebras.

Findings

Connected permutation-invariant sets imply connected spectral sets.

Spectral cones in simple Euclidean Jordan algebras are irreducible.

Eigenvalue orbits are arcwise connected in simple Euclidean Jordan algebras.

Abstract

Let V be a Euclidean Jordan algebra of rank n. The eigenvalue map from V to R^n takes any element x in V to the vector of eigenvalues of x written in the decreasing order. A spectral set in V is the inverse image of a permutation set in R^n under the eigenvalue map. If the permutation set is also a convex cone, the spectral set is said to be a spectral cone. This paper deals with connectedness and arcwise connectedness properties of spectral sets. By relying on the result that in a simple Euclidean Jordan algebra, every eigenvalue orbit is arcwise connected, we show that if a permutation invariant set is connected (arcwise connected), then the corresponding spectral set is connected (respectively, arcwise connected). A related result is that in a simple Euclidean Jordan algebra, every pointed spectral cone is irreducible.