In this question which I flag it as a community wiki, I search for a big list of $C^{*}$ algebras(and a big list of criterions) which do not admit a non trivial idempotent $C^{*}-$morphism.

I know 2 cases: Simple $C^{*}$ algebras, $B(H)$.

In this question which I flag it as a community wiki, I search for a big list of $C^{*}$ algebras(and a big list of criterions) which do not admit a non trivial idempotent $C^{*}-$morphism.

I know 2 cases: Simple $C^{*}$ algebras, $B(H)$, where $H$ is a separable Hilbert space.

In this question which I flag it as a community wiki, I search for a big list of $C^{*}$ algebras(and a big list of criterions) which do not admit a non trivial idempotent $C^{*}-$morphism.

I know 3 cases: Simple $C^{*}$ algebras, $B(H)$ and $Hom (E,E)$ where $E$ is a complex vector bundle over a compact space $X$ which does not have a proper sub bundle.Ex: canonical bundle over Grassmanian

In this question which I flag it as a community wiki, I search for a big list of $C^{*}$ algebras(and a big list of criterions) which do not admit a non trivial idempotent $C^{*}-$morphism.

I know 2 cases: Simple $C^{*}$ algebras, $B(H)$.