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

proofsof 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?