The Local Index Formula in Noncommutative Geometry Revisited

Abstract

In this review we discuss the local index formula in noncommutative geomety from the viewpoint of two new proofs are partly inspired by the approach of Higson especially that in but they differ in several fundamental aspedcts, in particular they apply to semifinite spectral triples for a *s-subalgebra A of a general semifinite von Neumann algebra. Our proofs are novel even in the setting of the original theorem and reduce the hypotheses of the theorem to those necessary for its statement.These proofs rely on the introduction of a function valued cocycle which is 'almost' a (b, B)-cocycle in the cyclic cohomology of A. They do not need the 'discrete dimension spectrum' assumption of jthe original Connes-Moscovici proof only a much weaker condition on the analytic continuation of certain zeta functions, and this only for part of the statement.In this article we also explain the relationship of the pairing between k-theory amd semifinite spectral triples to KK-theory and the Kasparov product. This discussion shows that simifinite spectral triples are a specific kind of representative of a K K-class and the analytically defined index is compatible with the Kasparov product. All authors were supported by grants from ARC (Australia) and NSERC (Canada). Abstract In this review a we discuss the local index formula in noncommutative geometry (NCG) from the viewpoint of two new proofs that are given in [16, 17] and [18] respectively. These proofs are partly inspired by the approach of Higson [35], especially that in [18], but they differ in several fundamental aspects, in particular they apply to semifinite spectral triples for a * -subalgebra A of a general semifinite von Neumann algebra. Our proofs are novel even in the setting of the original theorem, and reduce the hypotheses of the theorem to those necessary for its statement.These proofs rely on the introduction of a function valued cocycle which is 'almost' a (b, B)-cocycle in the cyclic cohomology of A. They do not need the 'discrete dimension spectrum' assumption of the original Connes-Moscovici proof [25], only a much weaker condition on the analytic continuation of certain zeta functions, and this only for part of the statement.In this article we also explain the relationship of the pairing between Ktheory and semifinite spectral triples to KK-theory and the Kasparov product. This discussion shows that semifinite spectral triples are a specific kind of representative of a KK-class, and the analytically defined index is compatible with the Kasparov product.

Index pairings for Rn-actions and Rieffel deformations

“…where τ is the extension of τ to L A (X) defined by (28). If n is even then for each projection e ∈ C ∼ for the product × Θ one has…”

“…In order to show that a "local" formula such as ( 3) is available for each class [u] ∈ K 1 (A) we must find a dense * -subalgebra C of A such that the inclusion C ֒→ A induces an isomorphism on K-theory (briefly, we need to find a "local subalgebra" of A) and such that the formula (3) holds for all unitaries u over the unitization C ∼ . That is a highly nontrivial task, and at several places one needs to apply tools which were developed quite recently [27,28,29,30,23,24]. After much work we will find a "smoothly summable" spectral triple (C, H, / D) such that, in both even and odd dimensions n, a generalization of (3) works for matrices over C ∼ .…”

Intrinsic Chern-Connes Characters for Crossed Products by $\mathbb Z^d$

“…As for the Breuer-Fredholm index [11], the injectivity modulo path components present in 4.1 may be lost and the information about K 0 (A) group, that can be extracted using the " T -index, can be limited. One recalls that this is not the case for the classic Fredholm index.…”

“…In d = 2, this identity is due to Connes [12], who used it to compute the 2-dimensional Chern characters of the convolution algebras C ∞ c (R 2 ) and C ∞ c (SL(2, R)). The local index formula of [3,32] can be seen as a particularization of the generic Connes-Moscovici formula [14] and its later extensions [19,10,9,11,8]. A recent related work is [1], where local index formulas for Rieffel deformed crossed products are derived.…”