This question is motivated by Yemon Choi's answer here: Epimorphisms have dense range in TopHausGrp?

It's well-known that the category of *unital* commutative C*-algebras and $*$-homomorphisms is dual to the category of compact Hausdorff spaces and continuous maps. One finds that for all C*-algebras (with $*$-homomorphisms as morphisms) that monomorphisms are just injective maps, and epimorphisms are surjections (the latter point is non-trivial-- see the final paper which Yemon suggests in the link above).

Suppose instead we look at *locally* compact Hausdorff spaces, with continuous maps as morphisms. Then dually, we get all commutative C*-algebras, but now the notion of a $*$-homomorphism is too restrictive (it corresponds to proper continuous maps). Instead we say that a morphism between C*-algebras $A$ and $B$ is a non-degenerate $*$-homomorphism $A\rightarrow M(B)$ form $A$ to the multiplier algebra of $B$. Such a map extends uniquely to a strictly continuous $*$-homomoprhisms $M(A)\rightarrow M(B)$, and so we can compose such maps. Hence we get a category. A little checking shows that the full subcategory of commutative C*-algebras, with these morphisms, is now dual to the category of locally compact Hausdorff spaces with continuous maps. (I think Woronowicz was the first person to articulate this view).

For C*-algebras, with morphisms as arrows, what are epimorphisms and monomorphisms?

Restricting to the commutative case, we can instead look at locally compact Hausdorff spaces, and reverse the arrows. So working through, a monomorphism remains just an injective map; but I see no simple description of epimorphisms (at the level of algebras-- for spaces, it's just injective continuous maps).

*Edit:* Maybe this notion of "non-degenerate" is confusing. If $f:X\rightarrow Y$ is a continuous map between locally compact Hausdorff spaces, then we define $f_*:C_0(Y)\rightarrow C^b(X); a \mapsto a\circ f$. Notice that we really do need the codomain to be all bounded continuous functions-- but that's okay, as $C^b(X)$ is just the multiplier algebra of $C_0(X)$, and $f_*$ turns out to be non-degenerate. Conversely, every non-degenerate $*$-homomorphism $C_0(Y)\rightarrow C^b(X)$ arises in this way (but a general $*$-homomorphism $C_0(Y)\rightarrow C^b(X)$ can be much more complicated).

-homomorphisms between (nonunital) C-algebras to nondegenerate ones, i.e. those $f \colon A \to B$ for which $\{f(a)b \mid a \in A, b \in B\}$ is dense in $B$. So we really want to know what epimorphisms in this category are, which should hopefully make the question easier. – Chris Heunen Mar 16 '11 at 20:58doesn'twant to restrict only to proper continuous maps between LCH spaces, but instead consider all continuous maps between such? – Yemon Choi Mar 17 '11 at 1:09