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Commutator estimates in von Neumann algebras. (English. Russian original) Zbl 1306.47045
Funct. Anal. Appl. 47, No. 1, 62-63 (2013); translation from Funkts. Anal. Prilozh. 47, No. 1, 77-79 (2013).
Let $$LS(\mathcal {M})$$ be the set of all operators locally measurable with respect to a von Neumann algebra $$\mathcal {M}$$. The center of $$LS(\mathcal {M})$$ is denoted by $$Z(LS(\mathcal {M}))$$.
The authors assert that, for every self-adjoint operator $$a\in LS(\mathcal {M})$$,
1.
there exists an operator $$c_0=c^*_0\in Z(LS(\mathcal {M}))$$ such that, for every $$\varepsilon>0$$, there exists a unitary self-adjoint operator $$u_{\varepsilon}\in \mathcal {M}$$ for which $$|[a, u_{\varepsilon}]|\geq (1-\varepsilon)|a-c_0|$$.
2.
if $$\mathcal {M}$$ is a finite or purely infinite $$\sigma$$-finite von Neumann algebra, then there exists an operator $$c_0=c^*_0\in Z(LS(\mathcal {M}))$$ and a self-adjoint unitary operator $$u_0\in \mathcal {M}$$ such that $$|[a, u_0]|=u^*_0|a-c_0|u_0+|a-c_0|$$.
Note that no proofs are provided in this paper.
##### MSC:
 47B47 Commutators, derivations, elementary operators, etc. 46L57 Derivations, dissipations and positive semigroups in $$C^*$$-algebras 46L10 General theory of von Neumann algebras
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