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

Question

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.)

Answer 1

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