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Recently, examples of an index theory for KMS states of circle actions were discovered, [9,13]. We show that these examples are not isolated. Rather there is a general framework in which we use KMS states for circle actions on a C * -algebra A to construct Kasparov modules and semifinite spectral triples. By using a residue construction analogous to that used in the semifinite local index formula we associate to these triples a twisted cyclic cocycle on a dense subalgebra of A. This cocycle pairs with the equivariant KK-theory of the mapping cone algebra for the inclusion of the fixed point algebra of the circle action in A. The pairing is expressed in terms of spectral flow between a pair of unbounded self adjoint operators that are Fredholm in the semifinite sense. A novel aspect of our work is the discovery of an eta cocycle that forms a part of our twisted residue cocycle. To illustrate our theorems we observe firstly that they incorporate the results in [9, 13] as special cases. Next we use the Araki-Woods III λ representations of the Fermion algebra to show that there are examples which are not Cuntz-Krieger systems.

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AbstractRecently, examples of an index theory for KMS states of circle actions were discovered, [9,13]. We show that these examples are not isolated.

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confidence: 90%

Twisted cyclic theory, equivariant KK-theory and KMS states

Carey

Neshveyev

Nest

et al. 2011

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…We found that SU q (2) violates the 'full spectral subspace' condition which is defined in subsection 4.3 below. For SU (2) q , which admits both non-trivial traces and KMS states [20], one may use both the local index formula and the method described in this Section. The information that these yield differs in that the two methods 'see' different parts of the algebra.…”

Section: New Directionsmentioning

confidence: 99%

“…We remark that the content of [20] is the subject of K. Tong's recent thesis while the final example relies on some joint work of Consani-Marcolli [28,29,30]. 4.1.…”

Section: New Directionsmentioning

confidence: 99%

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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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“…This review article has been written to provide a summary of semifinite NCG so that recent new applications of this theory are accessible. Applications in a number of different areas are emerging (see for example [30]) including to a kind of index theory for KMS states on C * -algebras [6,7,15,20]. …”

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confidence: 99%

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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Spectral flow invariants and twisted cyclic theory for the Haar state on SUq(2)

Cited by 13 publications

References 29 publications

Twisted cyclic theory, equivariant KK-theory and KMS states

Twisted cyclic theory, equivariant KK-theory and KMS states

Semi-Finite Noncommutative Geometry and Some Applications

The Local Index Formula in Noncommutative Geometry Revisited

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