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

Abstract

A well-known theorem of Korovkin asserts that if $\{T_k\}$ is a sequence of positive linear transformations on $C[a,b]$ such that $T_k(h)\rightarrow h$ (in the sup-norm on $C[a,b]$) for all $h\in \{1,\phi,\phi^2\}$, where $\phi(t)=t$ on $[a,b]$, then $T_k(h)\rightarrow h$ for all $h\in C[a,b]$. In particular, if $T$ is a positive linear transformation on $C[a,b]$ such that $T(h)=h$ for all $h\in \{1,\phi,\phi^2\}$, 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 $h\in \{e,p,p^2\}$, 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 span of the Jordan frame corresponding to the spectral decomposition of $p$; consequently, if a positive linear transformation coincides with the Identity transformation on a Jordan frame, then it is doubly stochastic. We also present sequential and weak-majorization versions.