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2014-02-18T17:49:00Z

129.7.134.49:

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2014-02-18T15:42:50Z

129.7.134.50: /* Omnibus Tests in One Way Analysis of Variance */

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Bravais lattice

2013-10-28T16:22:24Z

129.7.134.103:

'''''Noncommutative topology''''' in [[mathematics]] is a term applied to the strictly [[C*-algebra]]ic part of the [[noncommutative geometry]] program. The program has its origins in the [[Gel'fand duality]] between the [[topology]] of [[locally compact]] spaces and the [[algebraic structure]] of [[commutative operation|commutative]] [[C*-algebra]]s.

Several [[topological]] properties can be formulated as properties for the [[C*-algebra]]s without making reference to [[commutative operation|commutativity]] or the underlying [[space]], and so have an immediate generalization.

Amongst these are [[compact space|compactness]] (being [[unital algebra|unital]]), [[dimension]] ([[real rank (C*-algebras)|real]] or [[stable rank]]), [[connected space|connectedness]] ([[projectionsless algebra|projectionless algebra]]) and [[K-theory]]. So we think of a noncommutative C*-algebra as the algebra of functions on a 'noncommutative space' which does not exist classically.

A major tool in the field is a [[functor#Bifunctors|bivariant]] version of K-theory called [[KK-theory]]. It has a composition product

<math>KK(A,B)\times KK(B,C)\rightarrow KK(A,C)</math>

of which the ring structure in ordinary K-theory is a special case. The product gives the structure of a [[Category (topology)|category]] to KK. It has been related to [[Correspondence (mathematics)|correspondences]] of algebraic varieties.<ref>[http://arxiv.org/abs/math.QA/0512138 math.QA/0512138]</ref>

== Notes ==
<references/>

{{DEFAULTSORT:Noncommutative Topology}}
[[Category:Banach algebras]]
[[Category:Topology]]

{{topology-stub}}