Honours thesis: Exact sequences in graded $KK$-theory for Cuntz-Pimsner algebras

TL;DR

This thesis extends the six-term exact sequence in graded KK-theory to include non-compact left actions, enabling the computation of KK-theory for row-infinite graph C*-algebras.

Contribution

It generalizes existing exact sequences in graded KK-theory to non-compact correspondences, facilitating new calculations for complex Cuntz-Pimsner algebras.

Findings

Extended the six-term exact sequence to non-compact cases

Computed graded KK-theory for row-infinite graphs

Developed foundational theory for advanced KK-theoretic calculations

Abstract

In this thesis we generalise the six-term exact sequence in graded $KK$-theory obtained in a paper of Kumjian, Pask and Sims (2017) to allow correspondences with non-compact left action. In particular, this allows us to compute the graded $KK$-theory of row-infinite graphs. We develop the theory necessary for following the arguments of Kumjian, Pask and Sims and of Pimsner (1997), with detailed sections on Hilbert modules, $C^*$-correspondences, Crossed products, Toeplitz algebras, Cuntz-Pimsner algebras and $KK$-theory.