# Toeplitz operators and pseudo-extensions

## TL;DR

This paper characterizes when a tuple of contractions admits a Toeplitz operator, links non-pureness to pseudo-extensions, and explores the structure of Toeplitz operator algebras, providing new insights and proofs.

## Contribution

It introduces a simple necessary and sufficient condition for Toeplitz operators in commuting contractions and establishes a commutant pseudo-extension theorem with a novel proof approach.

## Key findings

- A simple condition for Toeplitz operators in commuting contractions.
- Equivalence between non-pureness and pseudo-extensions to unitaries.
- A new proof of the existence of a canonical unitary pseudo-extension.

## Abstract

There are three main results in this paper. First, we find an easily computable and simple condition which is necessary and sufficient for a commuting tuple of contractions to possess a non-zero Toeplitz operator. This condition is just that the adjoint of the product of the contractions is not pure. On one hand this brings out the importance of the product of the contractions and on the other hand, the non-pureness turns out to be equivalent to the existence of a pseudo-extension to a tuple of commuting unitaries. The second main result is a commutant pseudo-extension theorem obtained by studying the unique canonical unitary pseudo-extension of a tuple of commuting contractions. The third one is about the $C^*$-algebra generated by the Toeplitz operators determined by a commuting tuple of contractions. With the help of a special completely positive map, a different proof of the existence of the unique canonical unitary pseudo-extension is given.