Multiplicativity factors for C-numerical radii

For matrices $\text{A, C?}\text{C}{}_{\mathrm{n}}\text{x}{}_{\mathrm{n}}$ , the C-numerical radius of A is the nonnegative quantity

$\text{r}{}_{\mathrm{c}}\text{(A)=max{|tr(CU}{}^1\text{AU)|:Uunitary}}$

. This generalizes the classical numerical radius r(A). It is known that r_c constitutes a norm on $\text{C}{}_{\mathrm{n}}\text{x}{}_{\mathrm{n}}$ if and only if C is nonscalar and trC≠0. For all such C we obtain multiplicativity factors for r_c, i.e., constant μ>0 for which μr_c is submultiplicative on $\text{C}{}_{\mathrm{n}}\text{x}{}_{\mathrm{n}}$.