# Extending a representation of a $C^*$-algebra to its unitization

I have a basic question about representations of non-unital $$C^*$$-algebras.

Let $$A$$ be a non-unital $$C^*$$-algebra, and $$\rho\colon A\to\mathcal{B}(H)$$ a $$*$$-representation of $$A$$ on some separable infinite-dimensional Hilbert space $$H$$.

Suppose that $$\rho$$ satisfies the following two properties:

1. It is non-degenerate, i.e. $$\rho(A)\cdot H$$ is dense in $$H$$;
2. For any $$a\neq 0$$ in $$A$$, $$\rho(a)$$ is not a compact operator on $$H$$.

Question: Can we extend $$\rho$$ to a representation of its unitization $$A^+$$ on $$\mathcal{B}(H)$$, for the same space $$H$$?

Thoughts: Clearly if $$\rho(A)$$ doesn't contain $$1_{\mathcal{B}(H)}$$ then it is possible to extend $$\rho$$ in this way. But it's not clear to me that $$1_{\mathcal{B}(H)}$$ can't be in the image of $$\rho$$. (I'm not sure if conditions 1 and 2 are required, but I've included them just in case they help.)

Just take $$\overline{\rho}: A^+\to B(H): (a,\lambda)\mapsto \rho(a)+\lambda 1_H.$$ No assumptions are required.