Journal of Operator Theory

Volume 93, Issue 1, Winter 2025 pp. 251-289.

Commutator estimates for normal operators in factors with applications to derivations

Authors: Aleksei F. Ber (1), Matthijs J. Borst (2), Fedor A. Sukochev (3)
Author institution: (1) Department of Mathematics, National University of Uzbekistan, Vuzgorodok, Tashkent 100174, Uzbekistan
(2) Delft Institute of Applied Mathematics, Delft university of Technology, Delft, P.O. Box 5031 2600 GA, The Netherlands
(3) School of Mathematics and Statistics, University of New South Wales, Kensington, NSW 2052, Australia

Summary: For a normal measurable operator $a$ affiliated with a von Neumann factor $\mathcal{M}$ we show that if $\mathcal{M}$ is infinite, then there is $\lambda_0\in \mathbb{C}$ so that for $\varepsilon>0$ there are $u_{\varepsilon}=u_{\varepsilon}^*$ , $v_{\varepsilon}\in \mathcal{U}(\mathcal{M} )$ with $v_\varepsilon|[a,u_\varepsilon]|v_\varepsilon^*\geqslant (1-\varepsilon)(|a-\lambda_0\textbf{1}|+u_\varepsilon|a-\lambda_0\textbf{1}|u_\varepsilon).$ If $\mathcal{M}$ is finite, then there is $\lambda_0\in\mathbb{C}$ and $u,v\in\mathcal{U}(\mathcal{M} )$ so that $v|[a,u]|v^*\geqslant \frac{\sqrt{3}}{2}(|a-\lambda_0\textbf{1}|+u|a-\lambda_0\textbf{1}|u^*).$ These bounds are optimal for infinite factors, II $_1$ -factors and some I $_n$ -factors. Furthermore, for finite factors applying $\|\cdot\|_{1}$ -norms to the inequality provides estimates on the norm of the inner derivation $\delta_{a}:\mathcal{M} \to L_1(\mathcal{M} ,\tau)$ associated to $a$ . While by\break \cite[Theorem 1.1]{BBS} it is known for finite factors and self-adjoint $a\in L_1(\mathcal{M} ,\tau)$ that $\|\delta_{a}\|_{ \to L_1( ,\tau)} = 2\min_{z\in \mathbb{C}}\|a-z\|_{1}$ , we present concrete examples of finite factors and normal operators $a\in \mathcal{M}$ for which this fails.

DOI: http://dx.doi.org/10.7900/jot.2023apr21.2442
Keywords: von Neumann factors, operator inequalities, derivations, commutators

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