noncommutative topology

1 Noncommutative Topology

Noncommutative topologyMathworldPlanetmathPlanetmathPlanetmath is basically the theory of C*-algebrasPlanetmathPlanetmath ( But why the name noncommutative topology then?

It turns out that commutativePlanetmathPlanetmathPlanetmathPlanetmath C*-algebras and locally compact Hausdorff spacesPlanetmathPlanetmath ( 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 conceptsMathworldPlanetmath are of course present in noncommutative C*-algebras too. Thus, although noncommutative C*-algebras cannot be associated with ”standard” topological spacesMathworldPlanetmath, 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 C0(X), the algebra of complex-valued continuous functionsPlanetmathPlanetmath in X that vanish at . Notice that C0(X) is a commutative C*-algebra.

Conversely, given a commutative C*-algebra 𝒜, the Gelfand transform provides an isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath between 𝒜 and C0(X), for a suitable locally compact Hausdorff space X.

Furthermore, there is an equivalence ( between the categoryMathworldPlanetmath of locally compact Hausdorff spaces and the category of commutative C*-algebras. This is the content of the Gelfand-Naimark theoremMathworldPlanetmath.

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
topological space C*-algebra
proper map *-homomorphismMathworldPlanetmathPlanetmathPlanetmath
homeomorphism *-isomorphism
open subset ideal
closed subset quotientPlanetmathPlanetmath (
compact space algebra with unit
compactification unitizationMathworldPlanetmath
one-point compactification minimal unitization (
Stone-Cech compactification ( unitization
second countable separablePlanetmathPlanetmath
connected projectionless
connected componentsPlanetmathPlanetmath and topological sums projections
complement of singleton maximal ideal
Radon measure

3.1 Remarks:

1. Noncommutative topology can be considered as part of Algebraic Topology (NAAT).

2.A specialized form of noncommutative topology is generally known as Noncommutative Geometry ( 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