Representation of dense Subset

Question

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

Answer 1

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.

Answer 2

If I understand the question correctly, the answer is no. Let $\mathcal A=\mathbb C,\mathcal B=\mathbb Q[\pi]+i\mathbb Q[\pi]\simeq(\mathbb Q+i\mathbb Q)[x].$ Define $\varphi:\mathcal B\to \mathbb C,\ \varphi(p(\pi))=p(0),$ where $p\in(\mathbb Q+i\mathbb Q)[x].$ Then $\pi$ is a non-zero homomorphism. It cannot be extended to the identity mapping $\mathbb C\to\mathbb C.$