#### Document Type

Article

#### 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*⁻¹)

_{j}^{±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.

#### Disciplines

Algebra

#### Recommended Citation

L.G. Brown and C. Schochet, *K*₁ of the compact operators is zero, *Proceedings of the American Mathematical Society* **59(1)** (1976), 119-122.

## Comments

First published in the

Proceedings of the American Mathematical Society59(1)(1976, http://dx.doi.org/10.1090/S0002-9939-1976-0412863-0 ), published by the American Mathematical Society.