Commutator estimates for normal operators in factors with applications to derivations

Abstract

For a normal measurable operator $a$ affiliated with a von Neumann factor $\mathcal{M}$ we show: If $\mathcal{M}$ is infinite, then there is ${\lambda }_0\in \mathbb{C}$ so that for $\epsilon >0$ there are $u_{\epsilon }=u_{\epsilon }^{\ast }$, $v_{\epsilon }\in \mathcal{U}(\mathcal{M})$ with $$v_{\epsilon }|[a,u_{\epsilon }]|v_{\epsilon }^{\ast }\ge (1-\epsilon )(|a-{\lambda }_0\text{1}|+u_{\epsilon }|a-{\lambda }_0\text{1}|u_{\epsilon }).$$ If $\mathcal{M}$ is finite, then there is ${\lambda }_0\in \mathbb{C}$ and $u,v\in \mathcal{U}(\mathcal{M})$ so that $$v|[a,u]|v^{\ast }\ge \frac{\sqrt{3}}{2}(|a-{\lambda }_0\text{1}|+u|a-{\lambda }_0\text{1}|u^{\ast }).$$ These bounds are optimal for infinite factors, II${}_1$-factors and some I${}_n$-factors. Furthermore, for finite factors applying $\Vert \cdot {\Vert }_1$-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 [3,Theorem 1.1] it is known for finite factors and self-adjoint $a\in L_1(\mathcal{M},\tau )$ that $\Vert {\delta }_a{\Vert }_{\mathcal{M}\to L_1(\mathcal{M},\tau )}=2{\min }_{z\in \mathbb{C}}\Vert a-z{\Vert }_1$, we present concrete examples of finite factors $\mathcal{M}$ and normal operators $a\in \mathcal{M}$ for which this fails.