Abstract:
Let the von Neumann algebra ${\mathcal M}$ of operators act in the Hilbert space $\mathcal{H}$,
$\tau$ – exact normal semifinite trace on
$\mathcal{M}$, $S(\mathcal{M}, \tau )$ – ${}^*$-algebra of $\tau$-measurable operators and
$ L_1(\mathcal{M},\tau)$ is the Banach space of $\tau$-integrable operators.
If $P, Q \in S(\mathcal{M}, \tau )^{\text{id}}$ and
$P-Q\in L_1(\mathcal{M},\tau)$, then $\tau (P-Q)\in \mathbb{R}$.
In particular, if $A=A^3\in L_1(\mathcal{M}, \tau )$,
then $\tau (A)\in \mathbb{R}$.
Let $A, B \in S(\mathcal{M}, \tau)$ be tripopotents.
If $A-B\in L_1(\mathcal{M}, \tau )$ and $A+B\in \mathcal{M}$,
then $\tau (A-B)\in \mathbb{R}$.
Let $P, Q \in S(\mathcal{M}, \tau )^{\text{id)}}$ with
$P-Q\in L_1(\mathcal{M},\tau)$ and $P Q \in \mathcal{M}$.
Then for all $n\in \mathbb{N}$ we have $(P-Q)^{2n+1}\in L_1(\mathcal{M},\tau)$ and
$\tau ((P-Q)^{2n+1})=\tau (P-Q)\in \mathbb{R}$.
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