Unitary normalizers in finite-dimensional inclusions

TL;DR

This paper introduces a new combinatorial invariant called the normalizer matrix to characterize regular inclusions of finite-dimensional von Neumann algebras, revealing their structure, decomposition, and depth properties.

Contribution

It provides a complete matrix-theoretic characterization of regular inclusions and links regularity to the existence of unitary orthonormal bases under spectral conditions.

Findings

Regular inclusions decompose into basic blocks.

Regularity is equivalent to the existence of a unitary orthonormal basis under spectral conditions.

Regular inclusions are necessarily of depth two.

Abstract

We study regular inclusions of finite-dimensional von Neumann algebras from a matrix-theoretic perspective. To this end, we introduce a new combinatorial invariant of an inclusion, called the normalizer matrix, which encodes the structure of the normalizer purely at the level of the inclusion matrix. Using this invariant, we obtain a complete characterization of regular inclusions of finite-dimensional von Neumann algebras. As consequences, we show that every regular inclusion decomposes into finite direct sums and tensor products of basic building blocks, and that regular inclusions are necessarily of depth two. We further investigate the existence of unitary orthonormal bases in the sense of Pimsner-Popa and prove that, under a natural spectral condition, regularity is equivalent to the existence of a unitary orthonormal basis contained in the normalizer. These results provide a unified description of regularity, unitary bases, and depth through the normalizer matrix formalism.