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

Let V be a Euclidean Jordan algebra of rank n. A set E in V is said to be a spectral set if there exists a permutation invariant set Q in Rn such that E=λ−1(Q), where λ:V→Rn is the eigenvalue map that takes x∈V to λ(x) (the vector of eigenvalues of x written in the decreasing order). If the above Q is also a convex cone, we say that E is 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 [x]:={y:λ(y)=λ(x)} is arcwise connected, we show that if a permutation invariant set Q is connected (arcwise connected), then λ−1(Q) is connected (respectively, arcwise connected). A related result is that in a simple Euclidean Jordan algebra, every pointed spectral cone is irreducible.