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).

hopfian spaces, as in en.wikipedia.org/wiki/…; I'd guess that if Varadarajan looked for hopfian manifolds, other spaces should probably exist. – Mariano Suárez-Alvarez♦ Oct 16 '12 at 3:08wordto get someone started on looking for more detail. (I have either not heard of hopfian objects before, or completely forgotten about them.) – Yemon Choi Oct 16 '12 at 3:38