Let $A$ be a unital $\*$-algebra, and $B$ a unital $\*$-subalgebra of $A$. In addition, assume that there exists a $B$-$B$-sub-bimodule $C \subset A$, such that 
$$
A \simeq B \oplus C,
$$
where $\simeq$ means an isomorphism of $B$-$B$-bimodules. If $B$ is endowed with a pre-$C^\*$-norm $\|\*\|$, then is it always possible to $\|\*\|$ to a $C^\*$-norm for $A$?


P.S. If it helps, one can assume that $A$ admits a $C^\*$-norm which does not restrict to $\|\*\|$ on $B$.


Edit: I am not assuming that $B$ is complete with respect to $\|\*\\$.



Extending $C^*$-norms from $*$-subalgebras
Ignoring the P.S. the answer is an easy no: $A$ could be any unital $\*$-algebra with a codimension $1$ $\*$-ideal $C$. Then let $B = \mathbb{C}\cdot e$ where $e$ is the unit of $A$, so a $B$-$B$ bimodule is just a vector space. Of course $B$ is endowed with a C\*-norm but $A$ is arbitrary so it need not have a C\*-norm. The P.S. doesn't add anything because the C\*-norm on $B$, if it exists, is unique, so if $A$ has a C\*-norm then its restriction to $B$ must be $\|\cdot\|$.


Edit: if the C\*-norm on $B$ is only a pre-C\*-norm, as per the edit, then it need not be unique, but the answer is still no. Let $A\_0$ be the polynomials in $x$ with norm inherited from $C[0,1]$ and let $A = A\_0 \oplus \mathbb{C}$ ($l^\infty$ direct sum). Let $f = (f\_0, 2)$ where $f\_0$ is the polynomial $x \mapsto x$, let $B$ be the set of unital polynomials in $f$, and let $C$ be the $\mathbb{C}$ summand. Then $B$ is isomorphic to $A\_0$ as $\*$-algebras, but the norm on $A\_0$ transferred to $B$ has no extension to $A$. That's because $f -2$ is not invertible in $A$ so the norm of $f$ in $A$ must be at least $2$, whereas its norm in $B$ is $1$.


