Trace and Commutators of Measurable Operators Affiliated to a von Neumann Algebra

In this paper, we present new properties of the space $L_1(\mathcal{M},\tau)$ of integrable (with respect to the trace $\tau$) operators affiliated to a semifinite von Neumann algebra ${\mathcal M}$. For self-adjoint $\tau$-measurable operators $A$ and $B$, we find sufficient conditions of the $\tau$-integrability of the operator $\lambda I-AB$ and the real-valuedness of the trace $\tau(\lambda I- AB)$, where $\lambda \in \mathbb{R}$. Under these conditions, $[A,B]=AB-BA\in L_1(\mathcal{M},\tau)$ and $\tau([A, B])=0$. For $\tau$-measurable operators $A$ and $B=B^2$, we find conditions that are sufficient for the validity of the relation $\tau([A,B])=0$. For an isometry $U\in\mathcal{M}$ and a nonnegative $\tau$-measurable operator $A$, we prove that $U-A \in L_1(\mathcal{M},\tau)$ if and only if $I-A, I-U \in L_1(\mathcal{M},\tau)$. For a $\tau$-measurable operator $A$, we present estimates of the trace of the autocommutator $[A^*,A]$. Let self-adjoint $\tau$-measurable operators $X\geq 0$ and $Y$ are such that $[X^{1/2}, Y X^{1/2}] \in L_1(\mathcal{M},\tau)$. Then $\tau ([X^{1/2}, Y X^{1/2}])=it$, where $t \in \mathbb{R}$ and $t=0$ for $XY \in L_1(\mathcal{M},\tau)$.