*-algebra, isomorphic to a C*-algebra A C-*-algebra A is *-isomorphic to a *-algebra B. 
Any examples when in this case B is not A C-*-algebra?
 A: No, $B$ needs not be a $C^*$-algebra in general. Just take $A=B$ with a $C^*$ norm on one side and a non $C^*$-norm on the other side.
But note the following more interesting fact.
If $A$ and $B$ are $*$-isomorphic $C^*$-algebras, then the $*$-homomorphism is automatically isometric.
This follows from the fact that 
$$
\|x\|^2=\|x^*x\|=\rho(x^*x)
$$
where $\rho(x^*x)$ is the spectral radius of $x^*x$, which is algebraic and does not depend on the $C^*$-norm.
A: The question isn't well-defined. A precise question would be: If $A$ is a $C^*$-algebra, $B$ is a $*$-algebra and the underlying $*$-algebra of $A$ is isomorphic to $B$, does it follow that $B$ is the underlying $*$-algebra of a $C^*$-algebra? And is this $C^*$-algebra isomorphic to $A$?
The answer is yes. If $\phi : B \to A$ is a $*$-isomorphism, then $||-|| \circ \phi$ defines a norm on $B$, and it is easily checked that this defines a $C^*$-algebra structure on $B$. By construction $\phi$ is an isometric isomorphism of $C^*$-algebras.
