Abstract
In this paper we show that the self-adjoint Fredholm operators in a type II∞ factor form a classifying space forK 1 (X) ⊗R for X a compact Hausdorff space. We also extend this result to the standard Hilbert module over a simple, purely infinite C*-algebra which is either unital or has a countable approximate identity consisting of projections.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
[APT] C.A. Akemann, G.K. Pedersen and J. TomiyamaMultipliers of C * algebras. J. Funct. Anal. (1973) 277–301.
[AS] M. Atiyah and I. SingerIndex theory for skew-adjoint Fredholm operators Publ. Math. IHES No 37 (1969) 305–326.
[BI] M. BreuerFredholm theories in Von Neumann algebras I and II Math Ann. 178(1968), Math Ann. 180(1969).
[B2] M. BreuerTheory of Fredholm operators relative to a Von Neumann algebra Rocky Mountam Math. J. 3 No. 3(1973)
[B3] M. BreuerOn the homotopy type of the group of regular elements of semifinite von Neumann algebras Math Ann. 185 (1970) 61–74
[B1] B. Blackadar K-theory for operator algebras
MSRI Publication No. 5 Springer Verlag (1986)
[CP] A.L. Carey and J. PhillipsAlgebras almost commuting with Clifford algebras in a type H ∞ factor K Theory 4, No. 5 (1991) 445–478.
[C] J. Cuntz,K-Theory for certain C * algebras. Ann. of Math. 113 (1981) 181–197.
[Hu] S.T. Hu Homotopy theory New York, Academic Press, 1959
[Mil] J. MilnorOn spaces having the homtopy type of a CW-complex Trans. Amer. Math Soc. (1958) 272–280
[Min1] J. MingoK-theory and multipliers of stable C * algebras. Trans. Amer. Math. Soc. 299, 1 (1987), 397–411.
[Min2] J. MingoK-Theory and multipliers of stable C *-algebras Ph.d Thesis, Dalhousie University Halifax, N.S. 1982
[Pa] R. PalaisHomotopy theory of Infinite Dimensional Manifolds Topology 5 (1966) 1–16
[Ph] J. PhillipsK-theory relative to a semifinite factor Indiana Univ. Math. J. (1990) 339–354
[P1] V.S. PereraReal valued spectral flow in a type II ∞ factor Preprint-1996
[P2] V.S. PereraHomotopy Groups of Self-Adjoint Fredholm operators in a Hilbert C *-module Preprint-1996
[R] I. RaeburnK-theory and K-homology relative to a II ∞ factor Proc. Amer. Math. Soc. Vol 71 (1978) 294–298
[Sp] E. Spanier Algebraic Topology McGraw-Hill 1966
[S] V.S. Sunder An invitation to Von Neumann algebras Springer Verlag
[T] J.L. TaylorBanach algebras and topology Algebras in Analysis (Birmingham 1973) Academic Press London
[W-O] N.E. Wegge-Olsen K theory and C*-algebras Oxford 1993.
[W] G.W. WhiteheadGeneralized homology theories Trans. Amer Math. Soc. 102 (1962) 277–303
[Zh1] S. ZhangA Riesz decomposition property and ideal structure of multiplier algebras J. Operator Theory 24 (1990) 209–225
[Zh2] S. ZhangK-theory and Homotopy of certain groups and Infinite Grassmanian spaces associated to C *-algebras Internat. Journal of Math.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Basu, D. A classifying space forK 1 (X, R) and extensions to Hilbert modules. Integr equ oper theory 31, 287–298 (1998). https://doi.org/10.1007/BF01195120
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01195120