The universal algebra generated by a power partial isometry

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.