noncommutative topology

1 Noncommutative Topology

Noncommutative topology is basically the theory of $C^{*}$ -algebras (http://planetmath.org/CAlgebra). But why the name noncommutative topology then?

It turns out that commutative $C^{*}$ -algebras andlocally compact Hausdorff spaces (http://planetmath.org/LocallyCompactHausdorffSpace) are one and the same ”thing” (this will be explained further ahead). Every commutative $C^{*}$ -algebra corresponds to a locally compact Hausdorff space and vice-versa and there is a correspondence between topological properties of spaces and $C^{*}$ -algebraic properties (see the noncommutative topology dictionary below).

The $C^{*}$ -algebraic properties and concepts are of course present in noncommutative $C^{*}$ -algebras too. Thus, although noncommutative $C^{*}$ -algebras cannot be associated with ”standard” topological spaces, all the topological/ $C^{*}$ 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 $X$ , all of its topological properties are encoded in $C_{0}(X)$ , the algebra of complex-valued continuous functions in $X$ that vanish at . Notice that $C_{0}(X)$ is a commutative $C^{*}$ -algebra.

Conversely, given a commutative $C^{*}$ -algebra $\\mathcal{A}$ , the Gelfand transform provides an isomorphism between $\\mathcal{A}$ and $C_{0}(X)$ , for a suitable locally compact Hausdorff space $X$ .

Furthermore, there is an equivalence (http://planetmath.org/EquivalenceOfCategories) between the category of locally compact Hausdorff spaces and the category of commutative $C^{*}$ -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 $C^{*}$ -algebras are the same thing. The other reason is the correspondence between topological and $C^{*}$ -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. $\\,$

3.1 Remarks:

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 asNoncommutative Geometry (http://planetmath.org/NoncommutativeGeometry) and has been introduced and developed by Professor Alain Connes (Field Medialist in 1982 and Crafoord Prize in 2001).

单词	NoncommutativeTopology
释义	noncommutative topology 1 Noncommutative Topology Noncommutative topology is basically the theory of $C^{}$ -algebras (http://planetmath.org/CAlgebra). But why the name noncommutative topology then? It turns out that commutative $C^{}$ -algebras andlocally compact Hausdorff spaces (http://planetmath.org/LocallyCompactHausdorffSpace) are one and the same ”thing” (this will be explained further ahead). Every commutative $C^{}$ -algebra corresponds to a locally compact Hausdorff space and vice-versa and there is a correspondence between topological properties of spaces and $C^{}$ -algebraic properties (see the noncommutative topology dictionary below). The $C^{}$ -algebraic properties and concepts are of course present in noncommutative $C^{}$ -algebras too. Thus, although noncommutative $C^{}$ -algebras cannot be associated with ”standard” topological spaces, all the topological/ $C^{}$ 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 $X$ , all of its topological properties are encoded in $C_{0}(X)$ , the algebra of complex-valued continuous functions in $X$ that vanish at . Notice that $C_{0}(X)$ is a commutative $C^{}$ -algebra. Conversely, given a commutative $C^{}$ -algebra $\\mathcal{A}$ , the Gelfand transform provides an isomorphism between $\\mathcal{A}$ and $C_{0}(X)$ , for a suitable locally compact Hausdorff space $X$ . Furthermore, there is an equivalence (http://planetmath.org/EquivalenceOfCategories) between the category of locally compact Hausdorff spaces and the category of commutative $C^{}$ -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 $C^{}$ -algebras are the same thing. The other reason is the correspondence between topological and $C^{*}$ -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. $\\,$ 3.1 Remarks: 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 asNoncommutative Geometry (http://planetmath.org/NoncommutativeGeometry) and has been introduced and developed by Professor Alain Connes (Field Medialist in 1982 and Crafoord Prize in 2001).
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