Abstract:
Let $\varphi$ be a trace on von Neumann algebra $\mathcal{M}$, $A, B\in \mathcal{M}$ and $\|B\|<1$, $[A, B]=AB-BA$. Then $ \varphi (|[A, B]|)\leq 2 \varphi (|A|)$. Let $\tau$ be a faithful normal semifinite trace on $\mathcal{M}$, $S(\mathcal{M}, \tau)$ be the $\ast$-algebra of all $\tau$-measurable operators. If $A\in L_2(\mathcal{M}, \tau )$ and $\mathrm{Re} A=\lambda |A|$ with $\lambda \in \{-1, 1\}$, then $A=\lambda |A|$. An operator $A\in L_2(\mathcal{M}, \tau )$ is Hermitian if and only if $\tau (A^2)=\tau (A^*A)$. Let positive operators $A,B\in S(\mathcal{M}, \tau)$ be invertible in $S(\mathcal{M}, \tau)$ and $ Y:=(A^{-1}-B^{-1})(A-B)$. If $Y, A^{1/2}YA^{-1/2}\in L_1(\mathcal{M},\tau)$, then $\tau (Y)\leq 0$. Let an operator $A\in S(\mathcal{M},\tau)$ be hyponormal and $A=B+\mathrm{i}C$ be its Cartesian decomposition. If 1) $BC\in L_1(\mathcal{M},\tau)$, or 2) $C=C^3\in \mathcal{M}$ and $[B, C]\in L_1(\mathcal{M},\tau)$, then $A$ is normal.
Keywords:Hilbert space, linear operator, von Neumann algebra, trace, hyponormal operator, inequality, projection.
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