**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?



Does there exist any massive proper $C^*$-subalgebra?
It was proved by Karl H. Hofmann and Karl-H.Neeb [here](https://doi.org/10.1007/BF01270691 "Hofmann, K.H., Neeb, K.H. Epimorphisms of C^*-algebras are surjective. Arch. Math 65, 134–137 (1995). zbMATH review at https://zbmath.org/?q=an:0829.46044") (see [here](https://arxiv.org/abs/funct-an/9405003) for the preprint), that epimorphisms in the category of $C^{\star}$-algebras are surjective.


