Classification of facial exposedness of completely positive cones over symmetric cones

Abstract: We classify the facial exposedness of completely positive cones over symmetric cones in terms of the rank of the associated Euclidean Jordan algebras. The completely positive cones are facially exposed when the rank is at most $2$, but are not facially exposed when the rank is at least $5$. Facial exposedness is not completely determined by the rank when the rank is $3$ or $4$, but we provide a characterization in each case. The resulting classification of facial exposedness in fact agrees with the corresponding classification of spectrahedrality for completely positive cones.