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Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

 
 

 

Factorizations of invertible operators and $ K$-theory of $ C^*$-algebras


Author: Shuang Zhang
Journal: Bull. Amer. Math. Soc. 28 (1993), 75-83
MSC: Primary 46L80; Secondary 19K33, 46L05, 46M20
DOI: https://doi.org/10.1090/S0273-0979-1993-00334-3
MathSciNet review: 1164064
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Abstract: Let $ \mathcal{A}$ be a unital $ {\text{C}}^{\ast}$-algebra. We describe K-skeleton factorizations of all invertible operators on a Hilbert $ {\text{C}}^{\ast}$-module $ \mathcal{H}_\mathcal{A}$, in particular on $ \mathcal{H}={l^2}$, with the Fredholm index as an invariant. We then outline the isomorphisms $ {K_0}(\mathcal{A}) \cong {\pi _{2k}}({[p]_0}) \cong {\pi _{2k}}({GL}_r^p(\mathcal{A}))$ and $ {{K}_{1}}(\mathcal{A})\cong {{\pi }_{2k+1}}({[p]_0})\cong {{\pi }_{2k+1}}({GL}_r^p(\mathcal{A}))$ for $ k \geq 0$, where $ {[p]_0}$ denotes the class of all compact perturbations of a projection p in the infinite Grassmann space $ {Gr}^{\infty}(\mathcal{A})$ and $ {GL}_r^p(\mathcal{A})$ stands for the group of all those invertible operators on $ {\mathcal{H}_\mathcal{A}}$ essentially commuting with p.


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DOI: https://doi.org/10.1090/S0273-0979-1993-00334-3
Article copyright: © Copyright 1993 American Mathematical Society

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