We generalise the local index formula of Connes and Moscovici to the case of spectral triples for a * -subalgebra A of a general semifinite von Neumann algebra. In this setting it gives a formula for spectral flow along a path joining an unbounded self-adjoint Breuer-Fredholm operator, affiliated to the von Neumann algebra, to a unitarily equivalent operator. Our proof is novel even in the setting of the original theorem and relies on the introduction of a function valued cocycle which is 'almost' a (b, B)-cocycle in the cyclic cohomology of A.
Spectral triples for nonunital algebras model locally compact spaces in noncommutative geometry. In the present text, we prove the local index formula for spectral triples over nonunital algebras, without the assumption of local units in our algebra. This formula has been successfully used to calculate index pairings in numerous noncommutative examples. The absence of any other effective method of investigating index problems in geometries that are genuinely noncommutative, particularly in the nonunital situation, was a primary motivation for this study and we illustrate this point with two examples in the text.In order to understand what is new in our approach in the commutative setting we prove an analogue of the Gromov-Lawson relative index formula (for Dirac type operators) for even dimensional manifolds with bounded geometry, without invoking compact supports. For odd dimensional manifolds our index formula appears to be completely new. As we prove our local index formula in the framework of semifinite noncommutative geometry we are also able to prove, for manifolds of bounded geometry, a version of Atiyah's L 2 -index Theorem for covering spaces. We also explain how to interpret the McKean-Singer formula in the nonunital case.In order to prove the local index formula, we develop an integration theory compatible with a refinement of the existing pseudodifferential calculus for spectral triples. We also clarify some aspects of index theory for nonunital algebras.
This paper is comprised of two related parts. First we discuss which k-graph algebras have faithful traces. We characterise the existence of a faithful semifinite lowersemicontinuous gauge-invariant trace on C * (Λ) in terms of the existence of a faithful graph trace on Λ.Second, 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.2. k-Graph C * -Algebras 2.1. Higher-rank graphs and their C * -algebras. In this subsection we outline the basic notation and definitions of k-graphs and their C * -algebras. We refer the reader to [RSY] for a more thorough account.Higher-rank graphs. Throughout this paper, we regard N k as a monoid under pointwise addition. We denote the usual generators of N k by e 1 , . . . , e k , and for n ∈ N k and 1 ≤ i ≤ k, we denote the i th coordinate of n by n i ∈ N; so n = n i e i . For m, n ∈ N k , we write m ≤ n if m i ≤ n i for all i. By m < n, we mean m ≤ n and m = n. We use m ∨ n and m ∧ n to denote, respectively, the coordinate-wise maximum and coordinate-wise minimum of m and n; so that m ∧ n ≤ m, n ≤ m ∨ n and these are respectively the greatest lower bound and least upper bound of m, n in N k . Definition 2.1 (Kumjian-Pask [KP]). A graph of rank k or k-graph is a pair (Λ, d) consisting of a countable category Λ and a degree functor d : Λ → N k which satisfy the following factorisation property: if λ ∈ Mor(Λ) satisfies d(λ) = m + n, then there are unique morphisms µ, ν ∈ Mor(Λ) such that d(µ) = m, d(ν) = n, and λ = µ • ν.The factorisation property ensures (see [KP]) that the identity morphisms of Λ are precisely the morphisms of degree 0; that is {id o : o ∈ Obj(Λ)} = d −1 (0). This means that we may identify each object with its identity morphism, and we do this henceforth. This done, we can regard Λ as consisting only of its morphisms, and we write λ ∈ Λ to mean λ ∈ Mor(Λ).Since we are thinking of Λ as a kind of graph, we write r and s for the codomain and domain maps of Λ respectively. We refer to elements of Λ as paths, and to the paths of degree 0 (which correspond to the objects of Λ as above) as vertices. Extending these conventions, we refer to the elements of Λ with minimal nonzero degree (that is d −1 ({e 1 , . . . , e k }) as edges.
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