Abstract. Let $A,B,$ and $C$ be unital $C^{\ast }$-algebras with $B$ injective. Let $C$ be a subalgebra of $A$ and $B$ with $I_{C}=I_{A}$ and $I_{C}=I_{B}$, let $M$ be a complex subspace of $A$ with $c_{1}Mc_{2}\subseteq M$ for all $c_{1},c_{2}\in C$, and let $L\colon M\rightarrow B $ be a $w_{\rho }$ completely bounded $C$-bihomomorphism. Then there exists a $C$-bihomomorphism extension $ \widetilde{L}\colon A\rightarrow B$ of $L$ with $\|\widetilde{L}\|_{W_{\rho cb}}=\|L\|_{W_{\rho cb}}$ $(0< \varrho\leq 2)$. Let $A_{i}$ be a unital $C^{\ast }$-algebra and $L_{i}\colon A_{i}\rightarrow B(H_{i})$ be a completely bounded map $(i=1,2)$. We provide $w_{\varrho }$ norms on $A_{1}\otimes_{\min }A_{2}$ and inequalities involving $\|L_{1}\otimes_{\min }L_{2}\|_{w_{\varrho }cb}$.

AMS Subject Classification (1991): 46L05, 46L10

Received September 25, 2006, and in revised form November 8, 2007. (Registered under 6013/2009.)