Commutators of compact operators

Abstract

In [4] Pearcy and Topping initiated the study of additive commutators of compact operators on Hilbert space and asked if each projection of rank one has this form. In [3] Brown and Schochet raised a more general question. Suppose H is an algebra of operators that contains the trace class operators and is commutative modulo the trace class operators. If A and B are elements of H and A is compact, then does the commutator [A,B] = AB – BA have trace 0? (This question arose from an attempt to generalize certain results obtained by Brown [2], Section 3, in the case where H is also a *-algebra.

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