Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-24T09:55:28.474Z Has data issue: false hasContentIssue false
  • English
  • Français

KK-Theory and Spectral Flow in von Neumann Algebras

Published online by Cambridge University Press:  04 April 2012

J. Kaad ,
R. Nest  and
A. Rennie
J. Kaad
Affiliation:
Institute for Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark, [email protected]
R. Nest
Affiliation:
Institute for Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark, [email protected]
A. Rennie
Affiliation:
Mathematical Sciences Institute, John Dedman Building, Australian National University, Acton 0200, ACT, Australia, [email protected]
Get access

Abstract

We present a definition of spectral flow for any norm closed ideal J in any von Neumann algebra N. Given a path of selfadjoint operators in N which are invertible in N/J, the spectral flow produces a class in Ko(J).

Given a semifinite spectral triple (A, H, D) relative to (N, τ) with A separable, we construct a class [D] ∈ KK1(A, K(N)). For a unitary u ∈ A, the von Neumann spectral flow between D and u*Du is equal to the Kasparov product [u]A[D], and is simply related to the numerical spectral flow, and a refined C*-spectral flow.

Keywords

KK-theoryspectral flowvon Neumann algebra
Type
Research Article
Information
Journal of K-Theory , Volume 10 , Issue 2 , October 2012 , pp. 241 - 277
Copyright
Copyright © ISOPP 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Baaj, S., Julg, P., Théorie bivariante de Kasparov et opérateurs non bornés dans les C*-modules hilbertiens, C. R. Acad. Sci. Paris 296 (1983), 139–149.Google Scholar
2.Benameur, M-T., Carey, A.L., Phillips, J., Rennie, A., Sukochev, F.A., Wojciechowski, K.P., An Analytic Approach to Spectral Flow in von Neumann Algebras, in ‘Analysis, Geometry and Topology of Elliptic Operators-Papers in Honour of K. P Wojciechowski’, World Scientific, 2006, 297–352.CrossRefGoogle Scholar
3.Blackadar, B., K-Theory for Operator Algebras, Mathematical Sciences Research Institute Publications, 1986, Springer-Verlag New York.Google Scholar
4.Booβ-Bavnbek, B., Lesch, M., Phillips, J.Unbounded Fredholm operators and spectral flow, Canad. J. Math. 57 (2005), 225–250.Google Scholar
5.Breuer, M., Fredholm theories in von Neumann algebras. I, Math. Ann. 178 (1968), 243–254.CrossRefGoogle Scholar
6.Breuer, M., Fredholm theories in von Neumann algebras. II, Math. Ann. 180 (1969), 313–325.Google Scholar
7.Connes, A., Noncoummutative Geometry, Academic Press, 1994.Google Scholar
8.Carey, A., Phillips, J., Unbounded Fredholm Modules and Spectral Flow, Can. J. Math. 50 (1998), 673–718.Google Scholar
9.Carey, A., Phillips, J., Spectral flow in ⊝-summable Fredholm modules, eta invariants and the JLO cocycle, K Theory 31 (2004), 135–194.CrossRefGoogle Scholar
10.Carey, A., Phillips, J., Rennie, A., Sukochev, F. A., The local index formula in semifinite von Neumann algebras I. Spectral flow, Adv. Math. 202 No. 2 (2006), 451–516.Google Scholar
11.Carey, A., Phillips, J., Rennie, A., Sukochev, F. A., The local index formula in semifinite von Neumann algebras II: the even case, Adv. Math. 202 No. 2 (2006), 517–554.Google Scholar
12.Dixmier, J., Les algèbres d'opérateurs dans l'espace hilbertien (Algèbres de von Neumann), Paris, Gauthiers-Villars, Editeur-Imprimeur-Libraire, 55, Quai Des Grands Augustins, 1957.Google Scholar
13.Fack, T., Sur la notion de valeur caractéristique, J. Operator Theory 7 (1982), 307–333.Google Scholar
14.Higson, N., Roe, J., Analytic K-Homology, Oxford University Press, (2000).Google Scholar
15.Kasparov, G. G., The operator K-functor and extensions of C*-algebras, Math. USSR Izv. 16 (1981), 513–572.Google Scholar
16.Lance, E. C., Hilbert C*-modules, London Mathematical Society Lecture Note Series 210, Cambridge University Press, 1995.Google Scholar
17.Meise, R., Vogt, D., Introduction to Functional Analysis, Oxford Science Publications, Clarendon Press, 1997.Google Scholar
18.Pask, D., Rennie, A., The Noncommutative Geometry of Graph C* -Algebras I: The Index Theorem, J. Funct. An. 233 (2006), 92–134.Google Scholar
19.Pask, D., Rennie, A., Sims, A., The Noncommutative Geometry of K-Graph C*-Algebras, J. K-Theory 1 (no. 2) (2008), 259–304.Google Scholar
20.Pedersen, G. K., C*-Algebras and Their Automorphism Groups, Academic Press, London, New York, San Francisco, 1979.Google Scholar
21.Wegge-Olsen, N.E., K-theory and C*-algebras, Oxford University Press, 1993.CrossRefGoogle Scholar
22.Phillips, J., Self-adjoint Fredholm operators and spectral flow, Canad. Math. Bull. 39 (1996), 460–467.Google Scholar
23.Phillips, J., Spectral flow in type I and type II factors-a new approach, Fields Institute Communications 17 (1997), 137–153.Google Scholar
24.Phillips, J., Raeburn, I. F., An index theorem for Toeplitz operators with noncommutative symbol space, J. Funct. Anal. 120 (1993), 239–263.CrossRefGoogle Scholar
25.Rennie, A., Smoothness and Locality for Nonunital Spectral Triples, K-Theory 28 (2003), 127–165.Google Scholar
26.Rørdam, M., Larsen, F., Laustsen, N.J., An Introduction to K-Theory of C*-Algebras, Cambridge University Press, 2000 London Mathematical Society Student Text 49.Google Scholar
27.Skandalis, G., On the group of extensions relative to a semifinite factor, J. Operator Theory 13 (1985), 255–263.Google Scholar
28.Wahl, C., On the Noncommutative Spectral Flow, J. Ramanujan Math. Soc. 22 (no. 2) (2007), 135–187.Google Scholar