Assume that $B$ is a $C^{*}$ subalgebra of $A$. We say $B$ is totally non hereditary subalgebra of $A$ if not only $B$ is not a hereditary subalgebra but also it is not isomorphic to any hereditary subalgebra of $A$.

Example: For every $C^{*}$ algebra $A$, $c(A)$ is a totally non hereditary subalgebra of $\ell^{\infty}(A)$. Here $c(A)$ and $\ell^{\infty}(A)$ are the algebra of all convergent and bounded sequence with $A$-elements, respectively.

There are some $C^{*}$ algebras which does not have such subalgebras. Example $\mathbb{C}^{n}$

To what extent all $C^{*}$ algebras without totally nonhereditary subalgebras are classified? What are some more examples(other than $\mathbb{C}^{n}$)?

In the topological language, somehow, we are interested in some non trvial example of compact Hausdorff space $X$ such that each quotient of $X$ is homeomorphic to an open subset of $X$.What is a totally nonhereditary subalgebra of $B(H)$ where $H$ is an infinite dimensional Hilbert space? What is an example of a simple $C^{*}$ algebra which contains a totally non hereditary sub $C^{*}$ algebra? Is there any reference about this concept?