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?



A property of compact topological space via certain $C^*$ embedding in operator algebras
If $A\subseteq B(H)$ is hereditary and has unit $p$, then we must have $A=pB(H)p\cong B(pH)$. So the answer is, such an embedding exists if and only if $A\cong B(H)$ for some $H$. In particular, the only commutative example is $A=\mathbb C$.


