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Algebras of Toeplitz Matrices with Quaternion Entries

Muhammad Ahsan Khan Ameur Yagoub

TL;DR

The paper tackles the noncommutative challenge of classifying maximal left algebras of quaternion Toeplitz matrices by introducing the G_pq[A] construction: a family of left algebras inside the quaternion Toeplitz space T_n[A] determined by a subalgebra A of the quaternions and a pair p,q in the commutant of A. It proves closure and structural properties, derives conditions for algebra equality and maximality (notably {p,q}' = A), and shows the framework encompasses key subalgebras such as quaternion circulants, upper triangular, and lower triangular cases. The results provide a concrete, scalable pathway toward a broader understanding of quaternion Toeplitz algebras and their maximal left ideals, with potential implications for noncommutative matrix theory and computational applications.

Abstract

The classification of maximal left algebras of quaternion Toeplitz matrices is a harder problem that has received little attention up to now. In this paper, we introduce certain families of maximal left algebras of Toeplitz matrices with entries from an algebra of quaternions that cover various classes of the left algebras of quaternion Toeplitz matrices.

Algebras of Toeplitz Matrices with Quaternion Entries

TL;DR

The paper tackles the noncommutative challenge of classifying maximal left algebras of quaternion Toeplitz matrices by introducing the G_pq[A] construction: a family of left algebras inside the quaternion Toeplitz space T_n[A] determined by a subalgebra A of the quaternions and a pair p,q in the commutant of A. It proves closure and structural properties, derives conditions for algebra equality and maximality (notably {p,q}' = A), and shows the framework encompasses key subalgebras such as quaternion circulants, upper triangular, and lower triangular cases. The results provide a concrete, scalable pathway toward a broader understanding of quaternion Toeplitz algebras and their maximal left ideals, with potential implications for noncommutative matrix theory and computational applications.

Abstract

The classification of maximal left algebras of quaternion Toeplitz matrices is a harder problem that has received little attention up to now. In this paper, we introduce certain families of maximal left algebras of Toeplitz matrices with entries from an algebra of quaternions that cover various classes of the left algebras of quaternion Toeplitz matrices.
Paper Structure (4 sections, 10 theorems, 45 equations)

This paper contains 4 sections, 10 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Quaternions and their Basic Algebraic Properties
  3. Toeplitz Matrices with Quaternion Entries
  4. Algebras of Toeplitz Matrices with Quaternions Entries

Key Result

Theorem 2.2

Let $\alpha=\alpha_0+\alpha_1i+\alpha_2j+\alpha_3k$ and $\beta=\beta_0+\beta_1i+\beta_2j+\beta_3k$ in $\mathbb{H}$, then

Theorems & Definitions (21)

  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • ...and 11 more