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 $\newcommand{\Cstar}{{\rm C}^*}\Cstar$-algebras with unital $*$-homomorphisms, are anti-equivalent. Thus one can "translate" properties about compact spaces over to $\Cstar$-algebras. This can lead to a sort of "dictionary", see for example page 3 of Várilly's book on noncommutative geometry (Google Books link).
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 $\Cstar$-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 $\{ \phi(a)b \mathbin{\colon} a\in A,b\in B \}$ is linearly dense in $B$. Then the category of commutative $\Cstar$-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?
Best Answer
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 $A\to B$
0'') Continuous maps $\longleftrightarrow$ non-generate C*-homomorphisms $A\to M(B)$
1) compact $\longleftrightarrow$ unital
1') $\sigma$-compact $\longleftrightarrow$ has a countable approx. unit ($\iff$ has a strictly positive element)
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 $K_0$
15) second countable $\longleftrightarrow$ separable w.r.t. the C*-norm
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.