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Let V be a Euclidean Jordan algebra of rank n. The eigenvalue map from V toR^n takes any element x in V to the vector of eigenvalues of x written in thedecreasing order. A spectral set in V is the inverse image of a permutation setin 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 withconnectedness and arcwise connectedness properties of spectral sets. By relyingon the result that in a simple Euclidean Jordan algebra, every eigenvalue orbitis 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 simpleEuclidean Jordan algebra, every pointed spectral cone is irreducible.

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