  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.
