Crossed products by endomorphisms and reduction of relations in relative Cuntz–Pimsner algebras

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 allows us to establish a coincidence between relative Cuntz–Pimsner algebras and crossed products introduced.