  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 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.
