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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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