COMMUTATORS AND SQUARE-ZERO ELEMENTS IN BANACH ALGEBRAS

Abstract

Our initial result states that, in a certain class of Banach algebras which includes |$C^*$|-algebras and group algebras of locally compact groups, every commutator lies in the closed linear span of square-zero elements. The proof relies on the theory of Banach algebras with property |${\mathbb {B}}$|⁠. We then study several variations and extensions of this result. For instance, we show that in a von Neumann algebra every commutator is actually a finite sum of square-zero elements. We also consider the commutator ideal and the existence of some special square-zero elements.