We prove that for any two elements , in a factor , if commutes with all the unitary conjugates of , then either or is in . Then we obtain an equivalent condition for the situation that the -numerical radius is a weakly unitarily invariant norm on finite factors, and we also prove some inequalities on the -numerical radius on finite factors. As an application, we show that for an invertible operator in a finite factor , is in the weak operator closure of the set , where is a polynomial, is the -Aluthge transform of , and .
"A note on the -numerical radius and the -Aluthge transform in finite factors." Ann. Funct. Anal. 9 (4) 463 - 473, November 2018. https://doi.org/10.1215/20088752-2017-0061