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 Abstract: Let $\alpha:G \rightarrow G$ be an endomorphism of a discrete amenable group such that $[G:\alpha(G)]<\infty$. We study the structure of the $C^*$ algebra generated by the left convolution operators acting on the left regular representation space, along with the isometry of the space induced by the endomorphism. Author information: Department of Mathematics, University of California at Berkeley, Berkeley, CA 94720,USA ilan@math.berkeley.edu