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

Assume that \( \tau \) is a faithful normal semifinite trace on a von Neumann algebra \( {\mathcal{M}} \), \( I \) is the unit of \( \mathcal{M} \), \( S({\mathcal{M}},\tau) \) is the \( * \)-algebra of \( \tau \)-measurable operators, and \( L_{1}({\mathcal{M}},\tau) \) is the Banach space of \( \tau \)-integrable operators. We present a new proof of the following generalization of Putnam’s theorem (1951): No positive self-commutator \( [A^{*},A] \) with \( A\in S({\mathcal{M}},\tau) \) is invertible in \( {\mathcal{M}} \). If \( \tau \) is infinite then no positive self-commutator \( [A^{*},A] \) with \( A\in S({\mathcal{M}},\tau) \) can be of the form \( \lambda I+K \), where \( \lambda \) is a nonzero complex number and \( K \) is a \( \tau \)-compact operator. Given \( A,B\in S({\mathcal{M}},\tau) \) with \( [A,B]\in L_{1}({\mathcal{M}},\tau) \) we seek for the conditions that \( \tau([A,B])=0 \). If \( X\in S({\mathcal{M}},\tau) \) and \( Y=Y^{3}\in{\mathcal{M}} \) with \( [X,Y]\in L_{1}({\mathcal{M}},\tau) \) then \( \tau([X,Y])=0 \). If \( A^{2}=A\in S({\mathcal{M}},\tau) \) and \( [A^{*},A]\in L_{1}({\mathcal{M}},\tau) \) then \( \tau([A^{*},A])=0 \). If a partial isometry \( U \) lies in \( {\mathcal{M}} \) and \( U^{n}=0 \) for some integer \( n\geq 2 \) then \( U^{n-1} \) is a commutator and \( U^{n-1}\in L_{1}({\mathcal{M}},\tau) \) implies that \( \tau(U^{n-1})=0 \).