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

List

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

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

1) compact $\longleftrightarrow$ unital

2) point $\longleftrightarrow$ maximal ideal

3) closed embedding $\longleftrightarrow$ quotientclosed 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.
show/hide this revision's text 3 added 288 characters in body; added 17 characters in body; edited body

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

List

0) locally compact Hausdorff space $\longleftrightarrow$ commutative C*-algebra

1) compact $\longleftrightarrow$ unital

2) point $\longleftrightarrow$ maximal ideal

3) closed embedding $\longleftrightarrow$ quotient

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

5) homeomorphism $\longleftrightarrow$ automorphism

6) clopen subset $\longleftrightarrow$ projection

7) totally disconnected $\longleftrightarrow$ AF-algebra

7(AF = approximately finite dimensional)

8) One-point compactification $\longleftrightarrow$ unitalization

8

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

9

10) Borel measure $\longleftrightarrow$ positive functional

10

11) probability measure $\longleftrightarrow$ state

11

12) disjoint union $\longleftrightarrow$ product

12

13) product $\longleftrightarrow$ completed tensor product

13

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 4) see here. 6) This is covered in books about AF-algebras (Kedison?). 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). 89) ?, 9. 10) is the Riesz representation Theorem. 1011) follows from 9)10). 1112) asserts $C_0(X \coprod Y) = C_0(X) \times C_0(Y)$, which is trivial, 12. 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. 1314) is the Theorem of Serre-Swan.
show/hide this revision's text 2 added 86 characters in body

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

-

List-

0) locally compact Hausdorff space $\longleftrightarrow$ commutative C*-algebra

1) compact $\longleftrightarrow$ unital

2) point $\longleftrightarrow$ maximal ideal

3) closed embedding $\longleftrightarrow$ quotient

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

5) homeomorphism $\longleftrightarrow$ automorphism

6) totally disconnected $\longleftrightarrow$ AF-algebra

7) One-point compactification $\longleftrightarrow$ unitalization

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

9) Borel measure $\longleftrightarrow$ positive functional

10) probability measure $\longleftrightarrow$ state

11) disjoint union $\longleftrightarrow$ product

12) product $\longleftrightarrow$ completed tensor product

-

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

Proofs- [in construction...]

0),1),2),3),5) follow directly from Gelfand duality - details can be found, for example, in Murphey's book about C*-algebras. For 4) see here. 6) This is covered in books about AF-algebras (Kedison?). 7) follows from abstract nonsense and 1). 8) ?, 9) is the Riesz representation Theorem. 10) follows from 9). 11) asserts $C_0(X \coprod Y) = C_0(X) \times C_0(Y)$, which is trivial, 12) 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 Stone WeierstraßStone-Weierstraß. 13) is the Theorem of Serre-Swan.
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