𝐾₁ of the compact operators is zero

Brown, L. G.; Schochet, Claude

doi:10.1090/S0002-9939-1976-0412863-0

$K_{1}$ of the compact operators is zero
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by L. G. Brown and Claude Schochet PDF

Proc. Amer. Math. Soc. 59 (1976), 119-122 Request permission

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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