noncommutative topology

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

𝒜

, the

Gelfand transform

provides an

isomorphism

between

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.pdf

Nonabelian 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

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