Representation of dense Subset let $\mathcal B \subset \mathcal A$ a dense subset of a C*-algebra $\mathcal A$. 
I have a representation for $\mathcal B$. Can I then conclude that this is somehow also a representation for $\mathcal A$?
By a representation for $\mathcal B$, I mean that I have a Hilbertspace $H$ and a *-homomorphism $\pi: \mathcal B \rightarrow \mathcal L (H)$. With $\mathcal{L}(H)$ I mean the bounded operators on $H$. 
Cheers
Peter

Since your representation is a bounded linear map, it extends to $\mathcal A$. Then, using the density, you prove that it is also a $*$-homomorphism. Of course, as Yurii mentioned, this works if $\mathcal B$ is a subalgebra; if it is not, it is not really clear what "representation" would be. 
