Geodesic
Commutator Estimates in von Neumann Algebras
Funkcionalʹnyj analiz i ego priloženiâ, Tome 47 (2013) no. 1, pp. 77-79.

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Let $\mathcal{M}$ be a von Neumann algebra. For every self-adjoint locally measurable operator $a$, there exists a central self-adjoint locally measurable operator $c_0$ such that, given any $\varepsilon>0$, $|[a,u_\varepsilon]|\ge(1-\varepsilon)|a-c_0|$ for some unitary operator $u_\varepsilon\in\mathcal{M}$. In particular, every derivation $\delta\colon\mathcal{M}\to\mathcal{I}$ (where $\mathcal{I}$ is an ideal in $\mathcal{M}$) is inner, and $\delta=\delta_a$ for $a\in\mathcal{I}$.
Keywords: derivation, von Neumann algebra, measurable operator, symmetric operator ideal. 
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A. F. Ber; F. A. Sukochev. Commutator Estimates in von Neumann Algebras. Funkcionalʹnyj analiz i ego priloženiâ, Tome 47 (2013) no. 1, pp. 77-79. http://geodesic.mathdoc.fr/item/FAA_2013_47_1_a6/

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