1 Noncommutative Topology
It turns out that commutative -algebras and locally compact Hausdorff spaces (http://planetmath.org/LocallyCompactHausdorffSpace) 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 of mathematics was given the name ”noncommutative topology”.
In this , noncommutative topology can be seen as ”topology, but without spaces”.
2 The Commutative Case
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 . Notice that is a commutative -algebra.
Furthermore, there is an equivalence (http://planetmath.org/EquivalenceOfCategories) 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.
3 Noncommutative Topology 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 of ”noncommutative mathematics”, such as noncommutative measure theory.
|Topological properties and concepts||-algebraic properties and concepts|
|closed subset||quotient (http://planetmath.org/QuotientRing)|
|compact space||algebra with unit|
|one-point compactification||minimal unitization (http://planetmath.org/Unitization)|
|Stone-Cech compactification (http://planetmath.org/StoneVCechCompactification)||unitization|
|connected components and topological sums||projections|
|complement of singleton||maximal ideal|
1. Noncommutative topology can be considered as part of http://aux.planetphysics.us/files/books/167/Anatv1.pdfNonabelian Algebraic Topology (NAAT).
2.A specialized form of noncommutative topology is generally known as Noncommutative Geometry (http://planetmath.org/NoncommutativeGeometry) and has been introduced and developed by Professor Alain Connes (Field Medialist in 1982 and Crafoord Prize in 2001).
|Date of creation||2013-03-22 17:40:18|
|Last modified on||2013-03-22 17:40:18|
|Last modified by||asteroid (17536)|
|Defines||noncommutative topology dictionary|