<i>KK</i>-Theory and Spectral Flow in von Neumann Algebras

Kaad, Jens; Nest, Ryszard; Rennie, Adam

doi:10.1017/is012003003jkt185

Cited by 21 publications

(45 citation statements)

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“…Our view is that the natural setting for the theory of spectral triples is not the bounded operators on a Hilbert space with its ideal of compact operators, but the corresponding situation in a general semifinite von Neumann algebra. This view is supported by [24], and we will amplify on this viewpoint later using results from [37].…”

Section: 2mentioning

confidence: 68%

“…We remark that semifinite Kasparov modules and semifinite spectral triples provide information that is different from that of the standard theory [23]. We provide later in this article a summary of [37] where it is shown that a semifinite spectral triple for A represents an element of KK * (A, B), where B is the separable norm closed subalgebra of the compact operators in N generated by the resolvent of D and the commutators [F D , a] for a ∈ A where…”

Section: 2mentioning

confidence: 95%

“…There Connes and Cuntz show that cyclic n-cocycles for an appropriate algebra A are in oneto-one correspondence with traces on a certain ideal J n in the free product A * A. Assuming some positivity for this trace yields the same kind of Kasparov modules and semifinite Fredholm modules as are described in [37]. In other words, to realise all the cyclic cocycles for an algebra will, in general, necessitate considering semifinite Fredholm modules.…”

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confidence: 99%

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The Local Index Formula in Noncommutative Geometry Revisited

Carey¹,

Phillips²,

Rennie³

et al. 2013

Noncommutative Geometry and Physics 3

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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.

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Section: 2mentioning

confidence: 68%

Section: 2mentioning

confidence: 95%

mentioning

confidence: 99%

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The Local Index Formula in Noncommutative Geometry Revisited

Carey¹,

Phillips²,

Rennie³

et al. 2013

Noncommutative Geometry and Physics 3

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“…In [23], Connes and Cuntz show that cyclic n-cocycles for an appropriate algebra A are in one-to-one correspondence with traces on a certain ideal J n in the free product A * A. Assuming some positivity for this trace yields the same kind of 'semi-finite Kasparov modules' as are described in [37]. In other words, to realise all the cyclic cocycles for an algebra will, in general, necessitate considering semifinite Fredholm modules and semi-finite spectral triples.…”

Section: Introductionmentioning

confidence: 92%

“…This is explained by a result of Kaad-NestRennie [37]. They show that a semifinite spectral triple for A represents an element of KK 1 (A, J), where J is the σ-unital norm closed ideal of compact operators in N generated by the resolvent of D and the commutators […”

Section: 2mentioning

confidence: 99%

Semi-Finite Noncommutative Geometry and Some Applications

Carey

Phillips

Rennie

2013

Noncommutative Geometry and Physics 3

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These notes are a summary of talks given in Shonan, Japan in February 2008 with modifications from a later series of talks at the Hausdorff Institute for Mathematics in Bonn in July 2008 and at the Erwin Schr¨odinger Institute in October 2008. The intention is to give a short discussion of recent results in noncommutative geometry (NCG) where one extends the usual point of view of [22] by replacing the bounded operators B(H) on a Hilbert space H by certain sub-algebras; namely semi-finite von Neumann algebras. These are weakly closed subalgebras of the bounded operators on a Hilbert space that admit a faithful, normal semifinite trace. The exposition is partly historical and intended to explain where this idea came from and how it links to other developments in index theory and NCG going back over 20 years.

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Application of Semifinite Index Theory to Weak Topological Phases

Bourne

Schulz-Baldes

2018

MATRIX Book Series

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Recent work by Prodan and the second author showed that weak invariants of topological insulators can be described using Kasparov's KK-theory. In this note, a complementary description using semifinite index theory is given. This provides an alternative proof of the index formulae for weak complex topological phases using the semifinite local index formula. Real invariants and the bulk-boundary correspondence are also briefly considered.Date: October 8, 2018. 1 a twisted Z k -action and invariant trace. Therefore the main task here is the computation of the resolvent cocycle that represents the (semifinite) Chern character and its application to weak invariants. Furthermore, the bulk-boundary correspondence proved in [30,7] also carries over, which allows us to relate topological pairings of the system without edge to pairings concentrated on the boundary of the sample.Acknowledgements. We thank our collaborators, Alan Carey, Johannes Kellendonk, Emil Prodan and Adam Rennie, whose work this builds from. We also thank the anonymous referee, whose careful reading and suggestions have improved the manuscript. We are partially supported by the DFG grant SCHU-1358/6 and C. B. also thanks the Australian Mathematical Society and the Japan Society for the Promotion of Science for financial support.2. Review: Twisted crossed products and semifinite index theory 2.1. Preliminaries. Let us briefly recall the basics of Kasparov theory that are needed for this paper; a more comprehensive treatment can be found in [5,31]. Due to the anti-linear symmetries that exist in topological phases, both complex and real spaces and algebras are considered.Given a real or complex right-B C * -module E B , we will denote by (· | ·) B the B-valued inner-product and by End B (E) the adjointable endomorphisms on E with respect to this inner product. The rank-1 operators Θ e,f , e, f ∈ E B , are defined such thatThen End 00 B (E) denotes the span of such rank-1 operators. The compact operators on the module, End 0 B (E), is the norm closure of End 00 B (E). We will often work with Z 2 -graded algebras and spaces and denote by⊗ the graded tensor product (see [16, Section 2] and [5]). Also see [25, Chapter 9] for the basic theory of unbounded operators on C * -modules.Definition 1. Let A and B be Z 2 -graded real (resp. complex) C * -algebras. A real (complex) unbounded Kasparov module (A, π E B , D) is a Z 2 -graded real (complex) C * -module E B , a graded homomorphism π : A → End B (E), and an unbounded self-adjoint, regular and odd operator D such that for all a ∈ A ⊂ A, a dense * -subalgebra,

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KK-Theory and Spectral Flow in von Neumann Algebras

Cited by 21 publications

References 21 publications

The Local Index Formula in Noncommutative Geometry Revisited

The Local Index Formula in Noncommutative Geometry Revisited

Semi-Finite Noncommutative Geometry and Some Applications

Application of Semifinite Index Theory to Weak Topological Phases

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