Exel's crossed product and relative Cuntz-Pimsner algebras

We consider Exel's new construction of a crossed product of a C*-algebra A by an endomorphism alpha. We prove that this crossed product is universal for an appropriate family of eovariant 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.