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

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 $M$, $f(\bigtriangleup_{\lambda}(T))$ is in the weak operator closure of the set $\{\sum_{i=1}^{n}z_{i}U_{i}f(T)U_{i}^{*}|n\in\mathbb{N},(U_{i})_{1\leq i\leq n}\in \mathscr{U}(M),\sum_{i=1}^{n}|z_{i}|\leq 1\}$, where $f$ is a polynomial, $\bigtriangleup_{\lambda}(T)$ is the $\lambda$-Aluthge transform of $T$ and $0\leq\lambda \leq 1$.