Linear transformations under which the doubly stochastic matrices are invariant

Abstract:

Let $[{M_n}(C)]$ denote the set of linear maps from the $n \times n$ complex matrices into themselves and let ${\hat \Omega _n}$ denote the set of complex doubly stochastic matrices, i.e. complex matrices whose row and column sums are 1. If $F \in [{M_n}(C)]$ is such that $F({\hat \Omega _n}) \subseteq {\hat \Omega _n}$ and ${F^ \ast }({\hat \Omega _n}) \subseteq {\hat \Omega _n}$, then there exist ${A_i},{B_i},A$, and $B \in {\hat \Omega _n}$ such that \[ F(X) = \sum \limits _i {{A_i}X{B_i} + A{X^t}{J_n} + {J_n}{X^t}B - (1 + m){J_n}X{J_n}} \] for all $n \times n$ complex matrices $X$, where ${J_n}$ is the $n \times n$ matrix whose elements are each $1/n$ and where the superscript $t$ denotes transpose. $m$ denotes the number of the ${A_i}$ (or ${B_i}$).