The Partial-Isometric Crossed Products of c0 by the Forward and the Backward Shifts

Let (A, α) be a system consisting of a C*-algebra A and an extendible endomorphism α on A. We consider the partial-isometric crossed product A×αN generated by a copy of A and a power partial isometry. We show that for an extendible α-invariant ideal I in A, the quotient (A×αN)/(I×αN) of partial-isometric crossed products is isomorphic to the partial-isometric crossed product A/I× bar{α} N of the quotient algebra. Then we use this to give concrete descriptions of the partial-isometric crossed products of c0 by the forward shift and the backward shift.