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There are many different crossed products by an endomorphism of a C*-algebra,and constructions by Exel and Stacey have proved particularly useful. Here weshow that every Exel crossed product is isomorphic to a Stacey crossed product(though by a different endomorphism of a different C*-algebra), that everyStacey crossed product is an Exel crossed product, and answer some questionsraised by Ionescu and Muhly. While this manuscript is not yet in its finalform(s) several people have already used our results, so we are posting it now.

Exel and Stacey crossed products, and Cuntz-Pimsner algebras