Totally non hereditary $C^{*}$-subalgebras 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?

 A: So you want to consider a C*-algebra $A$ with the following property: Every sub-C*-algebra of $A$ is isomorphic to a hereditary sub-C*-algebra of $A$.
We can distinguish two cases:


*

*If $A$ is finite-dimensional, then $A$ has to be commutative. Indeed, assume $A\cong M_{k_1}\oplus\ldots\oplus M_{k_l}\oplus\mathbb{C}^n$ for $k_j\geq 2$ (and $j\geq 1$). Then $A$ has a commutative sub-C*-algebra of the form $\mathbb{C}^{k_1}\oplus\ldots\oplus \mathbb{C}^{k_l}\oplus\mathbb{C}^n$, but a commutative hereditary sub-C*-algebra of $A$ has at most $l+n$ points in its spectrum.

*If $A$ is infinite-dimensional, then it contains commutative sub-C*-algebras $C(X)$ where $X$ is any Peano continuum. Such a C*-algebra has to have `huge' primitive ideal space.
I don't think this property has been considered in the literature before.
Another remark: Every simple C*-algebra (not equal to $0$ or $\mathbb{C}$) contains a non-simple sub-C*-algebra. Every hereditary sub-C*-algebra of a simple C*-algebra is simple again. Therefore, every (nontrivial) simple C*-algebra contains a sub-C*-algebra with the condition you consider.
Similarly, $B(H)$ contains many sub-C*-algebras that are not isomorphic to a hereditary sub-C*-algebras of $B(H)$. This is because the primitive ideal space of $B(H)$ has only two points. Thus, any sub-C*-algebra $A$ of $B(H)$ such that the primitive ideal space of $A$ has at least three points will do.
