Brownlowe, Nathan and Raeburn, Iain, 2006, Exel's crossed product and relative Cuntz-Pimsner algebras, Cambridge Philosophical Society. Mathematical Proceedings, 141(3), 497-508.
We consider Exel's new construction of a crossed product of a C*-algebra A by an endomorphism α. We prove that this crossed product is universal for an appropriate family of covariant representations, and we show that it can be realised as a relative Cuntz-Pimsner algbera. We describe a necessary and sufficient condition for the canonical map from A into the crossed product to be injective, and present several examples to demonstrate the scope of this result. We also prove a gauge-invariant uniqueness theorem for the crossed product.