A NOTE ON THE C-NUMERICAL RADIUS AND THE λ-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 CI. Then we obtain an equivalent condition for the situation that the C-numerical radius omega(C)(.) 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 (Delta(lambda)(T)) is in the weak operator closure of the set {Sigma(n)(i)=z(i )U(i) f (T)U-i*vertical bar n is an element of N, (U-i)(1 <= i <= n) is an element of u (M), Sigma(n)(i)=vertical bar z(i)vertical bar <= 1}, where f is a polynomial, Delta(lambda)(T) is the lambda-Aluthge transform of T, and 0 <= lambda <= 1.