IUMJ

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

Authors: Kate Juschenko

Issue: Volume 60 (2011), Issue 4, 1171-1176

Abstract:

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.