The Noncommutative Geometry of k-graph C*-Algebras

2005/12/19 by David Pask, Pask, David, Adam Rennie +3
Mathematics · Physics and Astronomy · #46L05 #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #K-Theory and Homology (math.KT) #Noncommutative and Quantum Gravity Theories #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.math/0512438

openalex publication_date 2005/12/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper is comprised of two related parts. First we discuss which k-graph algebras have faithful gauge invariant traces, where the gauge action of \T^k is the canonical one. We give a sufficient condition for the existence of such a trace, identify the C*-algebras of k-graphs satisfying this condition up to Morita equivalence, and compute their K-theory. For k-graphs with faithful gauge invariant trace, we construct a smooth (k,∞)-summable semifinite spectral triple. We use the semifinite local index theorem to compute the pairing with K-theory. This numerical pairing can be obtained by applying the trace to a KK-pairing with values in the K-theory of the fixed point algebra of the \T^k action. As with graph algebras, the index pairing is an invariant for a finer structure than the isomorphism class of the algebra.

Related

• The Noncommutative Geometry of Graph C^*-Algebras I: The Index Theorem
• Noncommutative manifolds from graph and k-graph C^*-algebras
• Twisted k-graph algebras associated to Bratteli diagrams
• The ideal structure of the C*-algebras of infinite graphs
• Semifinite spectral triples associated with graph C*-algebras