Return to Question

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 (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 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 $\{ \phi(a)b \mid a\in A,b\in B \}$ 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?

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?

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?

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 (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 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 $\{ \phi(a)b \mid a\in A,b\in B \}$ 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?

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?

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?