Abstract. Let $A$ be a unital $C^*$-algebra and let $L\colon A\rightarrow B(H)$ be a linear map. Applying Proposition 2.2 and Theorem 2.4 of the paper, we can define operator radii $w_{\rho }(a)$ and induced completely bounded norms $\|L\| _{w_{\rho }cb}$, where $1\leq\rho \leq2$. When $\rho =2$, we prove that $$\|L\| _{wcb}=\inf\left\{\|\phi\| :\pmatrix{\phi & L \cr L^* & \phi} \hbox{ is completely positive}\right\}.$$ In general, we have the inequalities $|L|\leq\|L\| _{cb}\leq\|L\| _{w_{\rho}cb}\leq\|L\|_{wcb}\leq2|L|$, where $|L|=\inf\{\|\phi\|:\phi\pm\mathop{\rm Re}\alpha L$ is completely positive for all $|\alpha|=1\}$. We also give equivalent conditions for $\|L\| _{wcb}\leq1$ and extend the main Theorem of T. Ando and K. Okubo [1, p. 183] from $n\times n$ complex matrices to $n\times n$ matrices of operators on some Hilbert space.

AMS Subject Classification (1991): 46L05

Received February 4, 1999, and in revised form August 26, 1999. (Registered under 2733/2009.)