The Trace and Integrable Commutators of the Measurable Operators Affiliated to a Semifinite von Neumann Algebra.

Assume that is a faithful normal semifinite trace on a von Neumann algebra , is the unit of , is the -algebra of -measurable operators, and is the Banach space of -integrable operators. We present a new proof of the following generalization of Putnam's theorem (1951): No positive self-commutator with is invertible in . If is infinite then no positive self-commutator with can be of the form , where is a nonzero complex number and is a -compact operator. Given with we seek for the conditions that . If and with then . If and then . If a partial isometry lies in and for some integer then is a commutator and implies that . [ABSTRACT FROM AUTHOR]