A note on the $C$-numerical radius and the $\lambda$-Aluthge transform in finite factors

Abstract

We prove that for any two elements $A$, $B$ in a factor $M$, 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 $M$, $f({△}_{\lambda }(T))$ is in the weak operator closure of the set $\{{\sum }_{i=1}^n{z}_i{U}_{i}f(T)U_i^{\ast }|n\in \mathbb{N},(U_i{)}_{1\le i\le n}\in \mathcal{U}(M),{\sum }_{i=1}^n|z_i|\le 1\}$, where $f$ is a polynomial, ${△}_{\lambda }(T)$ is the $\lambda$-Aluthge transform of $T$ and $0\le \lambda \le 1$.

11pages. arXiv admin note: text overlap with arXiv:math/0512197 by other authors