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

TL;DR

This paper introduces a new construction of crossed products by endomorphisms of C*-algebras, highlighting their dependence on ideals, and establishes a reduction procedure linking relative Cuntz-Pimsner algebras with crossed products.

Contribution

It provides an explicit structure of crossed products by endomorphisms and a canonical reduction method for C*-correspondences to unify different C*-algebra constructions.

Findings

Explicit description of the internal structure of crossed products

Reduction procedure for C*-correspondences

Equivalence between relative Cuntz-Pimsner algebras and crossed products

Abstract

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.