A description of the logmodular subalgebras in the finite-dimensional $C^*$-algebras Kate Juschenko 47L5547A6747A20logmodular algebratriangular matricescompletely contractivehomomorphism 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 by V.I. Paulsen and M. Raghupathi in [V.I. Paulsen and M. Raghupathi, \textit{Representations of logmodular algebras}, Trans. Amer. Math. Soc. \textbf{363} (2011), no. 5, 2627--2640.] In particular, this shows that every unital contractive representation of a logmodular subalgebra of $M_n(\mathbb{C})$ is automatically completely contractive. Indiana University Mathematics Journal 2011 text pdf 10.1512/iumj.2011.60.4347 10.1512/iumj.2011.60.4347 en Indiana Univ. Math. J. 60 (2011) 1171 - 1176 state-of-the-art mathematics http://iumj.org/access/