The universal algebra generated by a power partial isometry

TL;DR

This paper characterizes the universal $C^*$-algebra generated by a power partial isometry, linking it to the algebra of finite sections for Toeplitz operators with continuous symbols, expanding understanding of operator algebra structures.

Contribution

It identifies the universal $C^*$-algebra generated by a PPI with a modified algebra of finite sections for Toeplitz operators, connecting abstract algebraic concepts with concrete operator theory.

Findings

The universal algebra generated by a PPI is explicitly characterized.

A connection is established between PPIs and Toeplitz operator algebras.

The algebra of finite sections for Toeplitz operators is shown to be universal for PPIs.

Abstract

A power partial isometry (PPI) is an element $v$ of a $C^*$-algebra with the property that every power $v^n$ is a partial isometry. The goal of this paper is to identify the universal $C^*$-algebra generated by a PPI with (a slight modification of) the algebra of the finite sections method for Toeplitz operators with continuous generating function, as first described by Albrecht B\"ottcher and Bernd Silbermann in 1983.