| 1. |
- Deeley, Robin J., et al.
(författare)
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The bordism group of unbounded KK-cycles
- 2018
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Ingår i: Journal of Topology and Analysis. - 1793-5253 .- 1793-7167. ; 10:2
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Tidskriftsartikel (refereegranskat)abstract
- © 2018 World Scientific Publishing Company We consider Hilsum’s notion of bordism as an equivalence relation on unbounded (Formula presented.)-cycles and study the equivalence classes. Upon fixing two (Formula presented.)-algebras, and a ∗-subalgebra dense in the first (Formula presented.)-algebra, a (Formula presented.)-graded abelian group is obtained; it maps to the Kasparov (Formula presented.)-group of the two (Formula presented.)-algebras via the bounded transform. We study properties of this map both in general and in specific examples. In particular, it is an isomorphism if the first (Formula presented.)-algebra is the complex numbers (i.e. for (Formula presented.)-theory) and is a split surjection if the first (Formula presented.)-algebra is the continuous functions on a compact manifold with boundary when one uses the Lipschitz functions as the dense ∗-subalgebra.
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| 2. |
- DEELEY, ROBIN J., et al.
(författare)
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Wieler solenoids, Cuntz–Pimsner algebras and K-theory
- 2018
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Ingår i: Ergodic Theory and Dynamical Systems. - : Cambridge University Press (CUP). - 0143-3857 .- 1469-4417. ; 38:8, s. 2942-2988
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Tidskriftsartikel (refereegranskat)abstract
- © Cambridge University Press, 2017 We study irreducible Smale spaces with totally disconnected stable sets and their associated (Formula presented.)-theoretic invariants. Such Smale spaces arise as Wieler solenoids, and we restrict to those arising from open surjections. The paper follows three converging tracks: one dynamical, one operator algebraic and one (Formula presented.)-theoretic. Using Wieler’s theorem, we characterize the unstable set of a finite set of periodic points as a locally trivial fibre bundle with discrete fibres over a compact space. This characterization gives us the tools to analyse an explicit groupoid Morita equivalence between the groupoids of Deaconu–Renault and Putnam–Spielberg, extending results of Thomsen. The Deaconu–Renault groupoid and the explicit Morita equivalence lead to a Cuntz–Pimsner model for the stable Ruelle algebra. The (Formula presented.)-theoretic invariants of Cuntz–Pimsner algebras are then studied using the Cuntz–Pimsner extension, for which we construct an unbounded representative. To elucidate the power of these constructions, we characterize the Kubo–Martin–Schwinger (KMS) weights on the stable Ruelle algebra of a Wieler solenoid. We conclude with several examples of Wieler solenoids, their associated algebras and spectral triples.
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| 3. |
- Forsyth, Iain, et al.
(författare)
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Boundaries, spectral triples and K-homology
- 2019
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Ingår i: Journal of Noncommutative Geometry. - 1661-6952 .- 1661-6960. ; 13:2, s. 407-472
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Tidskriftsartikel (refereegranskat)abstract
- © European Mathematical Society This paper extends the notion of a spectral triple to a relative spectral triple, an unbounded analogue of a relative Fredholm module for an ideal J G A. Examples include manifolds with boundary, manifolds with conical singularities, dimension drop algebras, -deformations and Cuntz–Pimsner algebras of vector bundles. The bounded transform of a relative spectral triple is a relative Fredholm module, making the image of a relative spectral triple under the boundary mapping in K-homology easy to compute. We introduce an additional operator called a Clifford normal with which a relative spectral triple can be doubled into a spectral triple. The Clifford normal also provides a boundary Hilbert space, a representation of the quotient algebra, a boundary Dirac operator and an analogue of the Calderon projection. In the examples this data does assemble to give a boundary spectral triple, though we can not prove this in general. When we do obtain a boundary spectral triple, we provide sufficient conditions for the boundary triple to represent the K-homological boundary. Thus we abstract the proof of Baum–Douglas–Taylor’s “boundary of Dirac is Dirac on the boundary” theorem into the realm of non-commutative geometry.
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| 4. |
- Goffeng, Magnus C H T, 1987, et al.
(författare)
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Spectral triples and finite summability on Cuntz-Krieger algebras
- 2015
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Ingår i: Documenta Mathematica. - 1431-0643 .- 1431-0635. ; 20, s. 89-170
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Tidskriftsartikel (refereegranskat)abstract
- We produce a variety of odd bounded Fredholm modules and odd spectral triples on Cuntz-Krieger algebras by means of realizing these algebras as “the algebra of functions on a non-commutative space” coming from a sub shift of finite type. We show that any odd K-homology class can be represented by such an odd bounded Fredholm module or odd spectral triple. The odd bounded Fredholm modules that are constructed are finitely summable. The spectral triples are theta-summable, although their phases will already on the level of analytic K-cycles be finitely summable bounded Fredholm modules. Using the unbounded Kasparov product, we exhibit a family of generalized spectral triples, related to work of Bellissard-Pearson, possessing mildly unbounded commutators, whilst still giving well defined K-homology classes.
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| 5. |
- Goffeng, Magnus C H T, 1987, et al.
(författare)
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Untwisting twisted spectral triples
- 2019
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Ingår i: International Journal of Mathematics. - 0129-167X. ; 30:14
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Tidskriftsartikel (refereegranskat)abstract
- We examine the index data associated to twisted spectral triples and higher order spectral triples. In particular, we show that a Lipschitz regular twisted spectral triple can always be "logarithmically dampened" through functional calculus, to obtain an ordinary (i.e. untwisted) spectral triple. The same procedure turns higher order spectral triples into spectral triples. We provide examples of highly regular twisted spectral triples with nontrivial index data for which Moscovici's ansatz for a twisted local index formula is identically zero.
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