A description of the logmodular subalgebras in the finite dimensional C^*-algebras

We show that every logmodular subalgebra of $M_n(\mathbb{C})$ is unitary equivalent to an algebra of block upper triangular matrices, which was conjectured in \cite{VM}. In particular, this shows that every unital contractive representation of a logmodular subalgebra of $M_n(\mathbb{C})$ is automatically completely contractive.