Trace class self-commutators

Abstract

This paper extends earlier results of Berger and Shaw to all W ∗ {W^\ast } algebras. The multiplicity of an operator in a W ∗ {W^\ast } algebra is defined in terms of the trace on the W ∗ {W^\ast } -algebra, and it is shown that if T T is a hyponormal operator in such an algebra, the trace of its self-commutator is bounded by this multiplicity times the area of the spectrum of T T , divided by π \pi .