A Complete Characterization of Regular Inclusions of Finite Dimensional $C^{\ast }$-algebras

Abstract

We give a complete characterization of regular (in the sense of Kumjian and Renault) unital inclusions of finite-dimensional $C^{\ast }$-algebras. For subalgebras ${⨁}_j({\mathbb{M}}_{d_j}(\mathbb{C})\otimes {\mathbb{I}}_{p_j})$ of ${\mathbb{M}}_n(\mathbb{C})$, we show that regularity depends only on equality of the multiplicities $p_j$, while unitary regularity---characterized recently by the first author and Silambarasan---additionally requires equality of the $d_j$; we recover the latter via a streamlined alternative proof. Extending this to inclusions of arbitrary finite-dimensional $C^{\ast }$-algebras, encoded by an inclusion matrix $\Lambda$, we show that regularity is equivalent to an explicit row/column condition on $\Lambda$---coinciding with the normalizer matrix introduced by the first author and Silambarasan ---so that the device used there to detect unitary regularity is shown to characterize regularity in general; unitary regularity is recovered by a further dimension-equality condition.

24 pages