Noncommutative geometry: Difference between revisions

Revision as of 23:40, 18 May 2004

Revision as of 09:05, 19 May 2004

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In [[mathematics]] and mathematical [[physics]], and in particular in [[functional analysis]], by analogy with the [[Gelfand representation]], which shows that [[commutative]] [[C-star-algebra|C* algebras]] are [[dual]] to [[locally compact]] [[Hausdorff space]]s, [[noncommutativenon-commutative]] C* algebras are often now called '''noncommutative spaces'''.

==Examples==

*The [[Symplectic space|symplectic]] [[phase space]] of [[Hamiltonian mechanics|classical mechanics]] is [[deform]]ed into a [[noncommutative]]non-commutative [[phase space]] generated by the position and momentem [[Operator (physics)|operators]].

Also, in analogy to the [[duality]] between [[affine scheme]]s and [[polynomial algebra]]s, we can also have noncommutative affine schemes.

For the duality between [[locally compact]] [[measure space]]s and [[commutative]] [[von Neumann algebra]]s, we call [[noncommutative]] [[von Neumann algebra]]s noncommutative''non-commutative measure spaces''.

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