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Equivariant $ K$-theory and the Chern character for discrete groups


Author: Efton Park
Journal: Proc. Amer. Math. Soc. 140 (2012), 745-747
MSC (2010): Primary 19L47, 47L65, 19K99
DOI: https://doi.org/10.1090/S0002-9939-2011-10912-2
Published electronically: June 8, 2011
MathSciNet review: 2869059
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Abstract: Let $ X$ be a compact Hausdorff space, let $ \Gamma$ be a discrete group that acts continuously on $ X$ from the right, define $ \widetilde{X} = \{(x,\gamma) \in X \times \Gamma : x\cdot\gamma= x\}$, and let $ \Gamma$ act on $ \widetilde{X}$ via the formula $ (x,\gamma)\cdot\alpha = (x\cdot\alpha, \alpha^{-1}\gamma\alpha)$. Results of P. Baum and A. Connes, along with facts about the Chern character, imply that $ K^i_\Gamma(X)$ and $ K^i(\widetilde{X}\slash\Gamma)$ are isomorphic up to torsion for $ i = 0, 1$. In this paper, we present an example where the groups $ K^i_\Gamma(X)$ and $ K^i(\widetilde{X}\slash\Gamma)$ are not isomorphic.


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Additional Information

Efton Park
Affiliation: Department of Mathematics, Box 298900, Texas Christian University, Fort Worth, Texas 76129
Email: e.park@tcu.edu

DOI: https://doi.org/10.1090/S0002-9939-2011-10912-2
Keywords: Equivariant $K$-theory, finite group actions, crossed products
Received by editor(s): June 14, 2010
Received by editor(s) in revised form: October 29, 2010, November 16, 2010, and November 23, 2010
Published electronically: June 8, 2011
Additional Notes: The author thanks the referee for helpful suggestions.
Communicated by: Brooke Shipley
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.