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 $L\colon M\to B$ be a $w_{2}$ completely bounded $C$-bihomomorphism. Then there exists a $C$-bihomomorphism extension $\widetilde{L}\colon A\to B$ of $L$ with $ \|\widetilde{L} \|_{w_{2}cb}= \|L \|_{w_{2}cb}.$ We also prove an extension theorem for a self-adjoint $w_{\rho }$ completely bounded $C$-bihomomorphism on a subspace of a unital $C^{\ast }$-algebra with $0< \rho\leq 2$.

AMS Subject Classification (1991): 46L05

Keyword(s): C, -bihomomorphism, w_{\rho }, contraction, w_{\rho }, completely bounded map

Received November 1, 2002. (Registered under 5807/2009.)