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noncommutative topology
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(Topic)
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Noncommutative topology is basically the theory of  -algebras. But why the name noncommutative topology then?
It turns out that commutative  -algebras and locally compact Hausdorff spaces are one and the same "thing" (this will be explained further ahead). Every commutative  -algebra corresponds to a locally compact Hausdorff space and vice-versa and there is a correspondence between topological properties of spaces and  -algebraic properties (see the
noncommutative topology dictionary below).
The  -algebraic properties and concepts are of course present in noncommutative  -algebras too. Thus, although noncommutative  -algebras cannot be associated with "standard" topological spaces, all the topological/ concepts are present. For this reason, this area of mathematics was given the name "noncommutative topology".
In this way, noncommutative topology can be seen as "topology, but without spaces".
Given a locally compact Hausdorff space  , all of its topological properties are encoded in  , the algebra of complex-valued continuous functions in  that vanish at infinity. Notice that  is a commutative  -algebra.
Conversely, given a commutative  -algebra
 , the Gelfand transform provides an isomorphism between
 and  , for a suitable locally compact Hausdorff space  .
Furthermore, there is an equivalence between the category of locally compact Hausdorff spaces and the category of commutative  -algebras. This is the content of the Gelfand-Naimark theorem.
This equivalence of categories is one of the reasons for saying that locally compact Hausdorff spaces and commutative  -algebras are the same thing. The other reason is the correspondence between topological and  -algebraic properties, present in the following dictionary.
We only provide a short list of easy-to-state concepts. Some correspondences of properties are technical and could not be easily stated here. Some of them originate new branches of "noncommutative mathematics", such as noncommutative measure theory. 
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"noncommutative topology" is owned by asteroid. [ full author list (2) ]
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Cross-references: Radon measure, maximal ideal, singleton, complement, projections, topological sums, connected components, connected, separable, second countable, one-point compactification, unitization, compactification, unit, compact, closed subset, ideal, open subset, homeomorphism, *-homomorphism, proper map, measure, equivalence of categories, Gelfand-Naimark theorem, category, isomorphism, Gelfand transform, vanish, continuous functions, algebra, topological spaces, noncommutative, locally compact Hausdorff space, commutative, Noncommutative Topology
This is version 7 of noncommutative topology, born on 2007-12-06, modified 2008-08-21.
Object id is 10110, canonical name is NoncommutativeTopology.
Accessed 548 times total.
Classification:
| AMS MSC: | 46L05 (Functional analysis :: Selfadjoint operator algebras :: General theory of $C^*$-algebras) | | | 46L85 (Functional analysis :: Selfadjoint operator algebras :: Noncommutative topology) | | | 54A99 (General topology :: Generalities :: Miscellaneous) |
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Pending Errata and Addenda
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