A note on the C-numerical radius and the λ-Aluthge transform in finite factors

Abstract

We prove that for any two elements $A$, $B$ in a factor $\mathcal{M}$, if $B$ commutes with all the unitary conjugates of $A$, then either $A$ or $B$ is in $\mathbb{C}I$. Then we obtain an equivalent condition for the situation that the $C$-numerical radius ${\omega }_C(\cdot )$ is a weakly unitarily invariant norm on finite factors, and we also prove some inequalities on the $C$-numerical radius on finite factors. As an application, we show that for an invertible operator $T$ in a finite factor $\mathcal{M}$, $f\left({△}_{\lambda }\right(T\left)\right)$ is in the weak operator closure of the set $\left\{{\sum }_{i=1}^n{z}_i{U}_{i}f\right(T)U_i^{\ast }\mid n\in \mathbb{N},(U_i{)}_{1\le i\le n}\in \mathcal{U}\left(\mathcal{M}\right),{\sum }_{i=1}^n\left|z_i\right|\le 1\}$, where $f$ is a polynomial, ${△}_{\lambda }\left(T\right)$ is the $\lambda$-Aluthge transform of $T$, and $0\le \lambda \le 1$.