The KK-Theory of Fundamental C*-Algebras

TL;DR

This paper establishes a long exact sequence in KK-theory for fundamental C*-algebras derived from graphs, unifying and extending previous results to include non GNS-faithful conditional expectations.

Contribution

It introduces a KK-theory long exact sequence for fundamental C*-algebras with non GNS-faithful expectations, generalizing prior work by multiple authors.

Findings

Proves KK-equivalence between full and vertex-reduced fundamental C*-algebras.

Unifies previous results by Cuntz, Pimsner, Germain, and Thomsen.

Extends results to non GNS-faithful conditional expectations.

Abstract

Given a graph of C*-algebras, we prove a long exact sequence in KK-theory for both the maximal and the vertex-reduced fundamental C*-algebras in the presence of possibly non GNS-faithful conditional expectations. We deduce from it the KK-equivalence between the full fundamental C*-algebra and the vertex-reduced fundamental C*-algebra even for non GNS-faithful conditional expectations. Our results unify, simplify and generalize all the previous results obtained before by Cuntz, Pimsner, Germain and Thomsen. It also generalizes the previous results of the authors on amalgamated free products.