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:
- It is non-degenerate, i.e. $\rho(A)\cdot H$ is dense in $H$;
- 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.)