Algebra Commons^™

K₁ Of The Compact Operators Is Zero, L. G. Brown, Claude Schochet

Mathematics Faculty Research Publications

We prove that K₁ 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_jB_jA_j⁻¹B_j⁻¹)^±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 …