Assume that $A$ is a unital $C^*$ algebra. Is there a $C^*$ embedding of $A$ in some $B(H)$ whose image is a hereditary $C^*$ subalgebra of $B(H)$?

If **not**, is the answer affirmative when $A$ is commutative?

If the answer of the later question is **affirmative**, we consider the following definition:

We say that a compact Hausdorff topological space has the property $P$ if for every Hilbert space $H$ and every two hereditary $C^*$ embeddings $\alpha, \beta: C(X) \to B(H)$, there is an automorphism $\phi$ of $B(H)$ with $\phi \circ \alpha=\beta$. Of course, this property is a topological invariant.

What are some examples of topological spaces which satisfy this property $P$? Does every finite set $X$ satisfy $P$? What are some other examples? Is this property preserved by disjoint union, product or wedge sum? Is this property identical to a well known classical property of topological spaces?