Minimizing the relative entropy in a face

Abstract

For a separating state ρ of a C ^*-algebra A, we give a limit formula for the minimal relative entropy S(ρ, ·) in any face, as well as for the unique minimizer. In terms of this minimum, we define a superadditive function $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\rho }$ on the faces of A. In the case of a W ^*-algebra and normal ρ, $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\rho }$ can be considered as function on the projection lattice of an abelian W ^*-subalgebra, which is dominated by $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\rho }$, is given by a normal positive, but not necessarily normalized linear functional on A. This functional is the unique solution of a minimal entropy problem.