# 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 |
---|---|

topological space | ${C}^{*}$-algebra |

proper map | *-homomorphism^{} |

homeomorphism | *-isomorphism |

open subset | ideal |

closed subset | quotient^{} (http://planetmath.org/QuotientRing) |

compact space | algebra with unit |

compactification | unitization^{} |

one-point compactification | minimal unitization (http://planetmath.org/Unitization) |

Stone-Cech compactification (http://planetmath.org/StoneVCechCompactification) | unitization |

second countable | separable^{} |

connected | projectionless |

connected components^{} and topological sums |
projections |

complement of singleton | maximal ideal |

Radon measure |

### 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 |