Reference request for translating from Top to C*-alg - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T20:25:04Z http://mathoverflow.net/feeds/question/82871 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/82871/reference-request-for-translating-from-top-to-c-alg Reference request for translating from Top to C*-alg Matthew Daws 2011-12-07T13:46:41Z 2013-03-09T13:24:56Z <p>Some recent questions on MO (for example, <a href="http://mathoverflow.net/questions/82708/do-subalgebras-of-cx-admit-a-description-in-terms-of-the-compact-hausdorff-spac" rel="nofollow">http://mathoverflow.net/questions/82708/do-subalgebras-of-cx-admit-a-description-in-terms-of-the-compact-hausdorff-spac</a>) have been about Gelfand duality-- namely, that the categories of compact Hausdorff spaces with continuous maps, and commutative unital C*-algebras with unital <code>$*$</code>-homomorphisms, are anti-equivalent. Thus one can "translate" properties about compact spaces over to C*-algebras. This can lead to a sort of "dictionary", see for example page 3 of Várilly's book on noncommutative geometry (which is <a href="http://books.google.co.uk/books?id=f5cAlaptxd0C&amp;lpg=PP1&amp;dq=non-commutative%20geometry&amp;pg=PA3#v=onepage&amp;q=non-commutative%20geometry&amp;f=false" rel="nofollow">http://books.google.co.uk/books?id=f5cAlaptxd0C&amp;lpg=PP1&amp;dq=non-commutative%20geometry&amp;pg=PA3#v=onepage&amp;q=non-commutative%20geometry&amp;f=false</a>).</p> <blockquote> <p>Does anyone know a reasonably definitive reference for <strong>proofs</strong> of such dictionaries, in a self-contained form??</p> </blockquote> <p>I'm guessing that perhaps such a thing doesn't exist, as these results are folklore (and are easy to prove really-- given the statement, the proofs often form nice exercises). As one is really just studying compact spaces via the <em>category</em> of compact spaces with continuous map, might there be a category theory book which is suitable?</p> <p>Actually, I am more interested in the non-unital case. Rather than working with proper maps, I instead want to follow Woronowicz. Define a "morphism" between C<code>$^*$</code>-algebras $A$ and $B$ to be a non-degenerate <code>$*$</code>-homomorphism $\phi:A\rightarrow M(B)$ from $A$ to the multiplier algebra of $B$, where "non-degenerate" means that <code>$\{ \phi(a)b : a\in A,b\in B \}$</code> is linearly dense in $B$. Then the category of commutative C<code>$^*$</code>-algebras and morphisms is anti-equivalent to the category of locally compact spaces and continuous maps. One can then form a similar dictionary-- but here I think the proofs can be a bit trickier (or maybe just they use slightly less standard topology).</p> <blockquote> <p>Does anyone know a reasonably definitive reference in this more general setting?</p> </blockquote> http://mathoverflow.net/questions/82871/reference-request-for-translating-from-top-to-c-alg/82960#82960 Answer by Martin Brandenburg for Reference request for translating from Top to C*-alg Martin Brandenburg 2011-12-08T12:20:51Z 2011-12-08T17:46:23Z <p>Let's make a list here. Everyone is invited to add and complete the list and the proofs.</p> <p><strong>List</strong> </p> <p>0) locally compact Hausdorff spaces $\longleftrightarrow$ commutative C*-algebras</p> <p>0') proper continuous maps $\longleftrightarrow$ non-degenerate C*-homomorphisms</p> <p>1) compact $\longleftrightarrow$ unital</p> <p>2) point $\longleftrightarrow$ maximal ideal</p> <p>3) closed embedding $\longleftrightarrow$ closed ideal</p> <p>4) surjection/injection $\longleftrightarrow$ injection/surjection</p> <p>5) homeomorphism $\longleftrightarrow$ automorphism</p> <p>6) clopen subset $\longleftrightarrow$ projection</p> <p>7) totally disconnected $\longleftrightarrow$ AF-algebra (AF = approximately finite dimensional)</p> <p>8) One-point compactification $\longleftrightarrow$ unitalization</p> <p>9) Stone-Cech compactification $\longleftrightarrow$ multiplier algebra</p> <p>10) Borel measure $\longleftrightarrow$ positive functional</p> <p>11) probability measure $\longleftrightarrow$ state</p> <p>12) disjoint union $\longleftrightarrow$ product</p> <p>13) product $\longleftrightarrow$ completed tensor product</p> <p>14) topological K-Theory $K^0$ $\longleftrightarrow$ algebraic K-theory $K_0$</p> <p><strong>Proofs</strong></p> <p>0),1),2),3),5) follow directly from Gelfand duality - details can be found, for example, in Murphey's book about C*-algebras. For 0'), see <a href="http://www.matheplanet.com/matheplanet/nuke/html/article.php?sid=1111" rel="nofollow">here</a> (I wrote this up because I didn't know any reference). A C*-homomorphism $A \to B$ is nondegenerate if the ideal generated by the image is dense. For 4) see <a href="http://mathoverflow.net/questions/82708/do-subalgebras-of-cx-admit-a-description-in-terms-of-the-compact-hausdorff-spac" rel="nofollow">here</a>. 6) is given by characteristic functions. A reference for 7) is Kenneth R. Davidson, C*-Algebras by Example, Theorem III.2.5. It is related to 6) because a commutative C*-algebra is AF iff it is separable and topologically generated by the projections. 8) follows from abstract nonsense and 1). 9) ?. 10) is the Riesz representation Theorem. 11) follows from 10). 12) asserts $C_0(X \coprod Y) = C_0(X) \times C_0(Y)$, which is trivial. 13) asserts that the canonical map $C_0(X) \hat{\otimes} C_0(Y) \to C_0(X \times Y)$ is an isomorphism - this follows from the Theorem of Stone-Weierstraß. 14) is the Theorem of Serre-Swan.</p> http://mathoverflow.net/questions/82871/reference-request-for-translating-from-top-to-c-alg/83018#83018 Answer by Amin for Reference request for translating from Top to C*-alg Amin 2011-12-09T01:23:07Z 2011-12-09T01:23:07Z <p>For 9 I have to say that I have also never seen a direct proof, but I think I remember some hints in Wegge-Olsen (in exercises of 'translation'); I can't garantee that it isn't the usual functorial argument tough...</p> http://mathoverflow.net/questions/82871/reference-request-for-translating-from-top-to-c-alg/83206#83206 Answer by Terry Loring for Reference request for translating from Top to C*-alg Terry Loring 2011-12-11T21:23:23Z 2013-03-09T13:24:56Z <p>Gert Pedersen wrote "In a careless moment a C*-algebraist might be quoted for saying that there is a covariant functor between the categories of commutative C*-algebras with morphisms and the category of locally compact Hausdorff spaces with continuous maps." (Morphisms of Extensions of C*-Algebras: Pushing Forward the Busby Invariant, by Eilers me and Pedersen.)</p> <p>Either of these correspondences is valid:</p> <ul> <li>List item</li> </ul> <p>proper continuous maps &lt;--> proper *-homomorphisms from A to B </p> <ul> <li>List item</li> </ul> <p>continuous maps &lt;--> nondegenerate *-homomorphisms from A to M(B).</p> <p>I misunderstood the exercise on Page 44 of Wegge-Olsen is wrong. This set of the discussion below.</p>