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 and locally 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. $\,$

Topological properties and concepts $C^{*}$-algebraic properties and concepts $C^{*}$-algebra *-homomorphism *-isomorphism ideal quotient (http://planetmath.org/QuotientRing) algebra with unit unitization minimal unitization (http://planetmath.org/Unitization) unitization separable projectionless projections maximal ideal

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 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).

 Title noncommutative topology Canonical name NoncommutativeTopology Date of creation 2013-03-22 17:40:18 Last modified on 2013-03-22 17:40:18 Owner asteroid (17536) Last modified by asteroid (17536) Numerical id 14 Author asteroid (17536) Entry type Topic Classification msc 54A99 Classification msc 46L85 Classification msc 46L05 Related topic GelfandTransform Related topic NoncommutativeGeometry Defines noncommutative topology dictionary Defines Noncommutative Geometry