A Complete Characterization of Regular Inclusions of Finite Dimensional $C^*$-algebras

Abstract

We give a complete characterization of regular (in the sense of Kumjian and Renault) unital inclusions of finite-dimensional $C^*$-algebras. For subalgebras $\bigoplus_j( \mathbb{M}_{d_j}(\mathbb{C}) \otimes \mathbb{I}_{p_j})$ 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 $C^*$-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.