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Definition 1: Suppose $B$ is a $C^* $-algebra. $A$ is massive $C^* $-subalgebra of $B$ iff
1. $A$ is a subalgebra of $B$;
2. for each irreducible representation $\pi$ of $B$ representation $\pi|_A$ is irreducible;
3. if representations $\pi$ and $\pi'$ aren't (unitary) equivalent then $\pi|_A$ and $\pi'|_A$ aren't equivalent too.

Definition 2: Suppose $B$ is a $C^* $-algebra. $A$ is massive $C^* $-subalgebra of $B$ iff inclusion $i\colon A\to B$ is epic in the category of $C^* $-algebras, i.e. for any $C^* $-algebra $D$ and any two homomorphisms $g_1,g_2\colon B\to D$ we have $g_1\circ i=g_2\circ i \Rightarrow g_1=g_2$.

It's not hard to prove, that these definitions are equivalent. But I don't know the answer to the following question:

Does there exist an example of massive proper $C^*$-subalgebra?

There is a theorem in the book of Dixmier "C*-algebras" (French original was published in 1969), that if $A$ is a massive $C^* $-subalgebra of postliminal $C^* $-algebra $B$ then $A=B$. What's for the general case? Is it an open problem?

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up vote 7 down vote accepted

It was proved by Karl H. Hofmann and Karl-H.Neeb here (see here for the preprint), that epimorphisms in the category of $C^{\star}$-algebras are surjective.

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During three days I were trying to prove this myself. It's amazing, that the proof is so simple (preprint is two pages long with nothing hard to understand). – Fiktor Oct 15 '10 at 17:48

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