Korovkin-type results and doubly stochastic transformations over Euclidean Jordan algebras

A well-known theorem of Korovkin asserts that if {T-k} is a sequence of positive linear transformations on C[a, b] such that T-k(h) -> h begin (in the sup-norm on C[a, b]) for all h is an element of{1,phi,phi(2)}, where phi(t)=t on [a,b], then T-k(h) -> h for all h is an element of C[a,b]. In particular, if T is a positive linear transformation on C[a,b]such that T(h)=h for all h is an element of{1,phi,phi(2)}, then T is the identity transformation. In this paper, we present some analogs of these results over Euclidean Jordan algebras. We show that if T is a positive linear transformation on a Euclidean Jordan algebra V such that T(h)=h for all h is an element of{e,p,p(2)}, where e is the unit element in V and p is an element of V with distinct eigenvalues, then T=T* =I(the identity transformation) on the sp ano f the Jordan frame corresponding to the spectral decomposition of p; consequently, if a positive linear transformation coincides with the identity transformation (more generally, an automorphism of V) on a Jordan frame, then it is doubly stochastic. We also present sequential and weak-majorization versions.