Graded $K$-theory and $K$-homology of relative Cuntz-Pimsner algebras and graph $C^*$-algebras

TL;DR

This paper develops exact sequences in KK-theory to compute graded K-theory and K-homology for relative Cuntz-Pimsner and graph C*-algebras, advancing understanding of their algebraic invariants.

Contribution

It introduces new exact sequences in KK-theory for graded relative Cuntz-Pimsner algebras and applies them to compute invariants of graph C*-algebras with edge labelings.

Findings

Calculated graded K-theory and K-homology for relative Cuntz-Krieger algebras.

Established exact sequences in KK-theory for graded Cuntz-Pimsner algebras.

Extended methods to graph C*-algebras with edge labelings.

Abstract

We establish exact sequences in $KK$-theory for graded relative Cuntz-Pimsner algebras associated to nondegenerate $C^*$-correspondences. We use this to calculate the graded $K$-theory and $K$-homology of relative Cuntz-Krieger algebras of directed graphs for gradings induced by $\{0,1\}$-valued labellings of their edge sets.