Pick a C*-algebra $A$ and call a (*-)endomorphism $\alpha:A\to A$ nontrivial if it is injective and $\alpha(A)\neq A$.
Question: Do there exist infinite dimensional C*-algebras with no nontrivial endomorphisms?
I'm particularly interested in the case of simple C*-algebras, but any example would do. In the commutative case the first part of this boils down to the following.
Question: Does there exist a locally compact Hausdorff space $X$ (with infinitely many points) for which every continuous surjection $X\to X$ is automatically injective?
I'm not sure such a space can exist, but it would necessarily be quite exotic: topological manifolds will not suffice (just work locally).