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

Abstract

Starting from an arbitrary endomorphism αof 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 orthogonal to kernel of 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.

The article is based on papers arXiv:math.OA/0703801 and arXiv:math.OA/0704.3811, and in essence forms their unification, refinement and development

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• Crossed products by endomorphisms of $C_0(X)$-algebras
• Exel's crossed product and crossed products by completely positive maps
• Topological freeness for $C^{\ast }$-correspondences
• Ideal structure of crossed products by endomorphisms via reversible extensions of $C^{\ast }$-dynamical systems
• Variational principles for spectral radius of weighted endomorphisms of $C(X,D)$
• Analysis of relationships between spectral potential of transfer operators, $t$-entropy, entropy and topological pressure
• Spectral triples on irreversible $C^{\ast }$-dynamical systems