[Paper Review] The Noncommutative Geometry of k-graph C*-Algebras

David Pask, Adam Rennie+1 more|ArXiv.org|Dec 19, 2005

Advanced Operator Algebra Research20 references4 citations

TL;DR

This paper establishes conditions for the existence of faithful semifinite traces on k-graph C*-algebras by linking them to graph traces on the underlying k-graph. It constructs (k,∞)-summable semifinite spectral triples for such algebras and computes the semifinite index pairing via KK-theory, showing it equals the KK-index with values in the K-theory of the fixed-point algebra under the T^k action, revealing a deep connection between semifinite index theory and KK-theory.

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 $(k,\infty)$-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.

Motivation & Objective

• To characterize when k-graph C*-algebras admit faithful, semifinite, lower-semicontinuous, gauge-invariant traces.
• To generalize the construction of semifinite spectral triples from graph algebras to k-graph algebras.
• To compute the semifinite index pairing using KK-theory and relate it to the K-theory of the fixed-point algebra under the T^k action.
• To explore the relationship between semifinite index theory and KK-theory in the context of k-graph algebras.

Proposed method

• Characterize the existence of a faithful gauge-invariant trace on C*(Λ) via the existence of a faithful graph trace on the k-graph Λ.
• Construct a Kasparov module for locally finite, locally convex k-graphs without sinks, which becomes a spectral triple when k is even.
• Use the T^k action on the k-graph C*-algebra to 'push forward' the Dirac operator from the torus T^k to define a Dirac operator on the k-graph algebra.
• Build a von Neumann algebra as part of the spectral triple that plays the role of the crossed product of the graph algebra by the T^k action.
• Apply the semifinite local index theorem to compute the index pairing with K-theory.
• Relate the index pairing to a KK-pairing with values in the K-theory of the fixed-point algebra of the T^k action.

Experimental results

Research questions

• RQ1 Under what conditions does a k-graph algebra admit a faithful, semifinite, gauge-invariant trace?
• RQ2 How can semifinite spectral triples be constructed for k-graph C*-algebras with such traces?
• RQ3 What is the relationship between the semifinite index pairing and KK-theory in this context?
• RQ4 How does the T^k action on a k-graph algebra facilitate the construction of a Dirac operator?
• RQ5 Can the index pairing computed via the semifinite local index theorem be expressed as a KK-index?

Key findings

• A k-graph algebra C*(Λ) admits a faithful, semifinite, lower-semicontinuous, gauge-invariant trace if and only if the k-graph Λ admits a faithful graph trace.
• For k-graphs with faithful graph traces, the paper constructs (k,∞)-summable semifinite spectral triples.
• The semifinite index pairing with K-theory equals the KK-index with values in the K-theory of the fixed-point algebra under the T^k action.
• The construction yields infinitely many examples of (semifinite) spectral triples of every integer dimension k ≥ 1.
• The k-graph algebras with faithful graph traces are Morita equivalent to direct sums of C*(G) for subgroups G ⊆ ℤ^k, hence to direct sums of continuous functions on tori of rank 0 to k.
• K-theory of such k-graph algebras is isomorphic to the direct sum of K-theory of tori of ranks l_v ≤ k, as computed via the decomposition into end components.