The notion of majorization of one self-adjoint $n \times n$ matrix by another is a very useful concept in linear algebra. For example, a classical theorem of Schur and Horn states that a diagonal matrix $D$ is majorized by a self-adjoint matrix $B$ if and only if a unitary conjugate of $B$ has the same diagonal as $D$. Some equivalent characterizations of $A$ being majorized by $B$ include $A$ being in the convex hull of the unitary orbit of $B$ (that is, there exists a mixed unitary quantum channel that maps $B$ to $A$), the eigenvalues of $A$ being controlled by the eigenvalues of $B$ via certain inequalities, tracial inequalities involving convex functions of $A$ and $B$, and doubly stochastic matrices relating the eigenvalues of $A$ and $B$.

In this talk, we will examine the notion of majorization in unital C$^*$-algebras. In particular, the closed convex hulls of the unitary orbits of a self-adjoint operator will be characterized, the possible values of the expectation of an operator onto a MASA in a II$_1$ factor will be described, and generalizations to (commuting) tuples of self-adjoint operators will be discussed.