  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
$0\leq\lambda \leq 1$.
