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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).

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Your spaces would be 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 3:08
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@Mariano: IMHO this is a great case of MO providing the right word to 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
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+1, I like it. But it took me a good minute of concentration to untangle the negations in the first question, so let me record the result of the untangling for anyone else as slow as me. The question asks whether there is an infinite-dimensional C*-algebra for which every injective endomorphism is surjective. –  Tom Leinster Oct 16 '12 at 17:02
    
@Tom: thanks for the clarification, I should have written it that way! –  Ollie Margetts Oct 17 '12 at 14:23
    
@Mariano: great, I'll see if there are references in and around Varadajan's paper. Unfortunately (or fortunately?) a search for cohopfian C*-algebras doesn't yield anything. –  Ollie Margetts Oct 17 '12 at 14:26
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2 Answers

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

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Thanks, David. I see that my memory for once was not faulty--Theorem 8 in Cook's paper gives what the OP wants. –  Bill Johnson Oct 16 '12 at 22:29
    
Great! I'll take a look (I find it a bit mind boggling that such a space can exist!) –  Ollie Margetts Oct 17 '12 at 14:33
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Look at $$ $$ Cook, H. Continua which admit only the identity mapping onto non-degenerate subcontinua. Fund. Math. 60 1967 241–249. $$ $$ The title and review give the main result, which does not give the example you seek. However, IIRC, Howard also constructed a non-degenerate compact metric space on which the only continuous self maps are the identity and constant functions. I can't access the mentioned paper to check if it is there.

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