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$ K\sb{1}$ of the compact operators is zero


Authors: L. G. Brown and Claude Schochet
Journal: Proc. Amer. Math. Soc. 59 (1976), 119-122
MSC: Primary 47B05; Secondary 58G15
DOI: https://doi.org/10.1090/S0002-9939-1976-0412863-0
MathSciNet review: 0412863
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Abstract: We prove that $ {K_1}$ of the compact operators is zero. This theorem has the following operator-theoretic formulation: any invertible operator of the form (identity) $ + $ (compact) is the product of (at most eight) multiplicative commutators $ {({A_j}{B_j}A_j^{ - 1}B_j^{ - 1})^{ \pm 1}}$, where each $ {B_j}$ is of the form (identity) $ + $ (compact). The proof uses results of L. G. Brown, R. G. Douglas, and P. A. Fillmore on essentially normal operators and a theorem of A. Brown and C. Pearcy on multiplicative commutators.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0412863-0
Keywords: Compact operator, essentially normal operator, algebraic $ K$-theory, extensions of $ {C^{\ast}}$-algebras
Article copyright: © Copyright 1976 American Mathematical Society

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