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| '''''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.
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| 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.
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| 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.
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| A major tool in the field is a [[functor#Bifunctors|bivariant]] version of K-theory called [[KK-theory]]. It has a composition product
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| <math>KK(A,B)\times KK(B,C)\rightarrow KK(A,C)</math> | |
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| 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>
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| == Notes ==
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| <references/>
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| {{DEFAULTSORT:Noncommutative Topology}}
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| [[Category:Banach algebras]]
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| [[Category:Topology]]
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| {{topology-stub}}
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Revision as of 22:14, 3 March 2014
I am Chas and was born on 9 February 1971. My hobbies are Australian Football League and Model Aircraft Hobbies.
my site ... Is No Nonsense Muscle Building Fake