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Unitary normalizers in finite-dimensional inclusions

Keshab Chandra Bakshi, Silambarasan C

TL;DR

The paper provides a complete finite-dimensional classification of regular inclusions $ e B esubset e A$ of von Neumann algebras via a new normalizer-matrix invariant, showing that regularity is equivalent to the inclusion matrix being a normalizer matrix with equal summand dimensions on each row-support. This yields a direct-sum and tensor-product decomposition into basic building blocks and proves that regular inclusions have depth two. Under a spectral condition on the inclusion matrix, regularity is equivalent to the existence of a unitary orthonormal basis contained in the normalizer, unifying regularity, unitary bases, and depth within the same combinatorial framework. The results connect finite-dimensional regularity with groupoid/weak-Hopf-algebra perspectives and have potential implications for quantum information (via unitary error bases) and C*-algebraic structure theory. Overall, the work provides a concrete, matrix-driven description of when regularity occurs and how to realize it via unitary bases inside the normalizer, plus a depth-two structural consequence.

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.

Unitary normalizers in finite-dimensional inclusions

TL;DR

The paper provides a complete finite-dimensional classification of regular inclusions of von Neumann algebras via a new normalizer-matrix invariant, showing that regularity is equivalent to the inclusion matrix being a normalizer matrix with equal summand dimensions on each row-support. This yields a direct-sum and tensor-product decomposition into basic building blocks and proves that regular inclusions have depth two. Under a spectral condition on the inclusion matrix, regularity is equivalent to the existence of a unitary orthonormal basis contained in the normalizer, unifying regularity, unitary bases, and depth within the same combinatorial framework. The results connect finite-dimensional regularity with groupoid/weak-Hopf-algebra perspectives and have potential implications for quantum information (via unitary error bases) and C*-algebraic structure theory. Overall, the work provides a concrete, matrix-driven description of when regularity occurs and how to realize it via unitary bases inside the normalizer, plus a depth-two structural consequence.

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.
Paper Structure (9 sections, 17 theorems, 149 equations)

This paper contains 9 sections, 17 theorems, 149 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Regular inclusions in finite dimension
  4. Normalizer matrices
  5. Normalizers of Subalgebras of Matrix Algebras
  6. Regularity of Inclusions in the General Case
  7. Unitary Orthonormal Bases and Depth Two Structure
  8. Unitary Orthonormal basis
  9. Depth of an inclusion

Key Result

Theorem 2.5

BB Let $(\mathcal{B}\subseteq \mathcal{A}, E)$ be a subalgebra system, with where $A$ is the inclusion matrix, and $m', n'$ are dimension vectors. Suppose $(\mathcal{B}\subseteq \mathcal{A}, E)$ admits a unitary orthonormal basis with $d$-unitary elements. Then Consequently, In particular, $m', n'$ are Perron-Frobenius eigenvectors of $A^tA$ and $AA^t$ respectively with eigenvalue $d$.

Theorems & Definitions (32)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • Definition 2.6
  • Proposition 3.1
  • Definition 3.2
  • Definition 3.3
  • Proposition 3.4
  • ...and 22 more