The paper presents a construction of the crossed product of a C * -algebra by an endomorphism generated by partial isometry.

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Starting from an arbitrary endomorphism α of a unital C * -algebra A we construct a crossed product. It is shown that the natural construction depends not only on the C * -dynamical system (A, α) but also on the choice of an ideal J orthogonal to ker α. The article gives an explicit description of the internal structure of this crossed product and, in particular, discusses the interrelation between relative Cuntz-Pimsner algebras and partial isometric crossed products. We present a canonical procedure that reduces any given C * -correspondence to the 'smallest' C * -correspondence yielding the same relative Cuntz-Pimsner algebra as the initial one. In the context of crossed products this reduction procedure corresponds to the reduction of C * -dynamical systems and allow us to establish a coincidence between relative Cuntz-Pimsner algebras and crossed products introduced.