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Some recent questions on MO (for example, Do subalgebras of C(X) admit a description in terms of the compact Hausdorff space X?) have been about Gelfand duality-- namely, that the categories of compact Hausdorff spaces with continuous maps, and commutative unital C*-algebras with unital $*$-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 http://books.google.co.uk/books?id=f5cAlaptxd0C&lpg=PP1&dq=non-commutative%20geometry&pg=PA3#v=onepage&q=non-commutative%20geometry&f=false).

Does anyone know a reasonably definitive reference for proofs of such dictionaries, in a self-contained form??

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 category of compact spaces with continuous map, might there be a category theory book which is suitable?

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$^*$-algebras $A$ and $B$ to be a non-degenerate $*$-homomorphism $\phi:A\rightarrow M(B)$ from $A$ to the multiplier algebra of $B$, where "non-degenerate" means that $\{ \phi(a)b : a\in A,b\in B \}$ is linearly dense in $B$. Then the category of commutative C$^*$-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).

Does anyone know a reasonably definitive reference in this more general setting?

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There is a third variation: the category of non-unital commutative $C^\ast$-algebras with $*$-homomorphisms is anti-equivalent to the category of pointed compact Hausdorff spaces with pointed continuous maps. I do not think that there is a good reference covering all of the dictionary, since - as you say - it is folklore. –  Andreas Thom Dec 7 '11 at 14:14
    
Your best bet might be someone's masters thesis, but unfortunately I have no ideas whose. Someone at Waterloo maybe? –  Yemon Choi Dec 7 '11 at 18:54
    
There is an old book in English published in Poland Banach spaces of continuous functions, vol. I, published before the Connes' revolution which is of some help books.google.com/books/about/… –  Zoran Skoda Dec 8 '11 at 14:55
    
Re: Zoran's suggestion - consulting my library's copy of Semadeni's book, it only seems to treat the unital case (sections 9.2.2, 10.2.4, 12.6.1 are the relevant ones) - I haven't yet found an explicit dictionary of the type Matt seems to be after. –  Yemon Choi Dec 8 '11 at 21:54

3 Answers 3

Let's make a list here. Everyone is invited to add and complete the list and the proofs.

List

0) locally compact Hausdorff spaces $\longleftrightarrow$ commutative C*-algebras

0') proper continuous maps $\longleftrightarrow$ non-degenerate C*-homomorphisms

1) compact $\longleftrightarrow$ unital

2) point $\longleftrightarrow$ maximal ideal

3) closed embedding $\longleftrightarrow$ closed ideal

4) surjection/injection $\longleftrightarrow$ injection/surjection

5) homeomorphism $\longleftrightarrow$ automorphism

6) clopen subset $\longleftrightarrow$ projection

7) totally disconnected $\longleftrightarrow$ AF-algebra (AF = approximately finite dimensional)

8) One-point compactification $\longleftrightarrow$ unitalization

9) Stone-Cech compactification $\longleftrightarrow$ multiplier algebra

10) Borel measure $\longleftrightarrow$ positive functional

11) probability measure $\longleftrightarrow$ state

12) disjoint union $\longleftrightarrow$ product

13) product $\longleftrightarrow$ completed tensor product

14) topological K-Theory $K^0$ $\longleftrightarrow$ algebraic K-theory $K_0$

Proofs

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 here (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 here. 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.

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@Martin: nice list! At some point your list will include $K^*(X)=K_*(C(X))$, which is not considered trivial by any standard (= the Serre-Swan theorem). (:-) –  Alain Valette Dec 8 '11 at 13:45
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Oh yes! I've included it. –  Martin Brandenburg Dec 8 '11 at 14:32
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I think it's worth pointing out that (4) becomes more subtle when the spaces are locally compact but not compact. In that case, a map $f: X \to Y$ induces an injection $C(Y) \to C(X)$ if and only if the range of $f$ is dense in $Y$. Of course if $X$ is compact then $f(X)$ is closed, so there's no difference for compact spaces. –  MTS Dec 8 '11 at 17:21
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@MTS: The correct maps between locally compact hausdorff spaces are here proper maps. These have closed image. I will add this to 0). –  Martin Brandenburg Dec 8 '11 at 17:39
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@Martin: I wanted to add sub-$C^*$-algebras to the list, but couldn't because your post is not CW. –  Rasmus Bentmann Jul 30 '12 at 9:31

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...

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If someone feels like filling the details, it's page 44 of K-theory and C* algebras...I've survived so far with the 'abstract non-sense'. –  Amin Dec 9 '11 at 1:31
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@Amin: See 3.12.6 in Pedersen's C*-algebra book. Everything in 9 lines (:-) –  Alain Valette Dec 9 '11 at 7:59
    
Thanks, that's nice ! –  Amin Dec 9 '11 at 10:40

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.)

Either of these correspondences is valid:

  • List item

proper continuous maps <--> proper *-homomorphisms from A to B

  • List item

continuous maps <--> nondegenerate *-homomorphisms from A to M(B).

I misunderstood the exercise on Page 44 of Wegge-Olsen is wrong. This set of the discussion below.

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Actually, I should have read Wegge-Olsen more closely-- that collection of exercises is really useful! When you get a chance, could you be more specific about which bit of the exercise you think "is wrong"...?? –  Matthew Daws Dec 12 '11 at 10:10
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Mea Culpa. The exercise on page 44 of Wegge-Olsen is correct. The exercise does not say that this functor is an equivalence of categories. It becomes an equivalence of categories if you consider only proper *-homomorphisms from A to B. By proper I mean it sends an approximate unit of A to an approximate unit of B. –  Terry Loring Dec 13 '11 at 0:24
    
@TerryLoring -- If the exercise you mentioned is in fact correct, could you edit your answer to reflect that, when you get a chance? It's a bit confusing to have the correction hidden in a comment... –  Vectornaut Feb 26 '13 at 18:57

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