Mono- and epi-morphisms for C*-algebras - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T03:59:55Z http://mathoverflow.net/feeds/question/58416 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/58416/mono-and-epi-morphisms-for-c-algebras Mono- and epi-morphisms for C*-algebras Matthew Daws 2011-03-14T12:02:14Z 2011-03-17T10:21:45Z <p>This question is motivated by Yemon Choi's answer here: <a href="http://mathoverflow.net/questions/56453/epimorphisms-have-dense-range-in-tophausgrp/56459#56459" rel="nofollow">http://mathoverflow.net/questions/56453/epimorphisms-have-dense-range-in-tophausgrp/56459#56459</a></p> <p>It's well-known that the category of <em>unital</em> commutative C*-algebras and <code>$*$</code>-homomorphisms is dual to the category of compact Hausdorff spaces and continuous maps. One finds that for all C*-algebras (with <code>$*$</code>-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).</p> <p>Suppose instead we look at <em>locally</em> compact Hausdorff spaces, with continuous maps as morphisms. Then dually, we get all commutative C*-algebras, but now the notion of a <code>$*$</code>-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 <code>$*$</code>-homomorphism $A\rightarrow M(B)$ form $A$ to the multiplier algebra of $B$. Such a map extends uniquely to a strictly continuous <code>$*$</code>-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).</p> <blockquote> <p>For C*-algebras, with morphisms as arrows, what are epimorphisms and monomorphisms?</p> </blockquote> <p>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).</p> <p><em>Edit:</em> 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 <code>$*$</code>-homomorphism $C_0(Y)\rightarrow C^b(X)$ arises in this way (but a general <code>$*$</code>-homomorphism $C_0(Y)\rightarrow C^b(X)$ can be much more complicated).</p> http://mathoverflow.net/questions/58416/mono-and-epi-morphisms-for-c-algebras/58445#58445 Answer by Chris Heunen for Mono- and epi-morphisms for C*-algebras Chris Heunen 2011-03-14T16:49:15Z 2011-03-14T16:49:15Z <p>I'm not sure why <code>*</code>-homomorphisms would be too restrictive a choice, but taking those as morphisms between C*-algebras as objects, the epimorphisms are precisely the surjective <code>*</code>-homomorphisms. This is proposition 2 in G. A. Reid's "Epimorphisms and surjectivity", Inventiones Mathematicae 9:295-307, 1970.</p>