Isometries of the Toeplitz Matrix Algebra

TL;DR

This paper characterizes the structure of isometries on the algebra of upper-triangular Toeplitz matrices, showing they are essentially unitary conjugations or conjugate-unitary conjugations, and classifies linear isometries as unitary similarities.

Contribution

It provides a complete description of continuous and linear isometries on Toeplitz matrix algebra, revealing their forms and algebraic properties.

Findings

Continuous multiplicative isometries are either unitary conjugations or conjugate-unitary conjugations.

Linear isometries are of the form A↦UAV with unitaries U,V.

Unital linear isometries are algebra homomorphisms.

Abstract

We study the structure of isometries defined on the algebra $\mathcal{A}$ of upper-triangular Toeplitz matrices. Our first result is that a continuous multiplicative isometry $\mathcal{A}\to M_n$ must be of the form either $A\mapsto UAU^*$ or $A\mapsto U\overline AU^*$, where $\overline A$ is the complex conjugation and $U$ is a unitary matrix. In our second result we use a range of ideas in operator theory and linear algebra to show that every linear isometry $\mathcal{A}\to M_n(\mathbb{C})$ is of the form $A\mapsto UAV$ where $U$ and $V$ are two unitary matrices. This implies, in particular, that every such an isometry is a complete isometry and that a unital linear isometry $\mathcal{A}\to M_n(\mathbb{C})$ is necessarily an algebra homomorphism.