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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

$K$-theory and $K$-homology relative to a $\textrm {II}_{\infty }$-factor


Author: Iain Raeburn
Journal: Proc. Amer. Math. Soc. 71 (1978), 294-298
MSC: Primary 55N15; Secondary 46L99, 46M20
DOI: https://doi.org/10.1090/S0002-9939-1978-0500509-4
MathSciNet review: 500509
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Abstract: Let X be a compact space and M be a factor of type $\mathrm {II}_\infty$ acting on a separable Hilbert space. Let ${K_M}(X)$ denote the Grothendieck group generated by the semigroup of isomorphism classes of M-vector bundles over X, and, if X is also metric, let $\operatorname {Ext}^M(X)$ denote the group of equivalence classes of extensions of $C(X)$ relative to M. We show that ${K_M}(X)$ is the direct sum of the even-dimensional Čech cohomology groups of X, and that $\operatorname {Ext}^M(X)$ is the direct product of the odd-dimensional Čech homology groups of X.


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Article copyright: © Copyright 1978 American Mathematical Society