  Starting from an arbitrary endomorphism \alpha 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,\alpha) but also on the choice of an
ideal orthogonal to kernel of \alpha. 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.
