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

TL;DR

This paper proves that all logmodular subalgebras of finite-dimensional complex matrices are unitarily equivalent to block upper triangular matrices, ensuring their unital contractive representations are automatically completely contractive.

Contribution

It confirms the conjecture that logmodular subalgebras in finite-dimensional matrix algebras are unitarily equivalent to block upper triangular matrices.

Findings

Logmodular subalgebras are unitarily equivalent to block upper triangular matrices.

Unital contractive representations of these subalgebras are automatically completely contractive.

The result applies to all such subalgebras in $M_n(\mathbb{C})$.

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 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.