TITLE: C*-algebras with no nontrivial endomorphisms
QUESTION [12 upvotes]: 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).

REPLY [3 votes]: Trying to add information to Bill Johnson's Answer, I'd just like to say that the paper he mentioned by Cook is available for free on-line.  Hopefully, below is a working link to the Cook paper in PDF format:
matwbn.icm.edu.pl/ksiazki/fm/fm60/fm60123.pdf
David Bernier