  Let $\mathcal A\subseteq \mat$ be a unital $*$-subalgebra of the algebra
$\mat$ of all $n\times n$ complex matrices and let $B$ be an hermitian matrix.
Let $\U_n(B)$ denote the unitary orbit of $B$ in $\mat$ and let $\mathcal
E_\mathcal A$ denote the trace preserving conditional expectation onto
$\mathcal A$. We give an spectral characterization of the set $$ \mathcal
E_\mathcal A(\U_n(B))=\{\mathcal E_\mathcal A(U^* B U): U\in \mat,\
\text{unitary matrix}\}.$$ We obtain a similar result for the contractive orbit
of a positive semi-definite matrix $B$. We then use these results to extend the
notions of majorization and submajorization between self-adjoint matrices to
spectral relations that come together with extended (non-commutative)
Schur-Horn type theorems.
