Editing Noncommutative geometry (section)

{{Short description|Branch of mathematics}}
'''Noncommutative geometry''' ('''NCG''') is a branch of [[mathematics]] concerned with a geometric approach to [[noncommutative algebra]]s, and with the construction of ''spaces'' that are locally presented by noncommutative algebras of functions, possibly in some generalized sense. A noncommutative algebra is an [[associative algebra]] in which the multiplication is not [[commutative]], that is, for which <math>xy</math> does not always equal <math>yx</math>; or more generally an [[algebraic structure]] in which one of the principal [[binary operation]]s is not commutative; one also allows additional structures, e.g. [[topology]] or [[norm (mathematics)|norm]], to be possibly carried by the noncommutative algebra of functions.

An approach giving deep insight about noncommutative spaces is through [[Operator algebra|operator algebras]], that is, algebras of [[Bounded linear operator|bounded linear operators]] on a [[Hilbert space]].{{sfn|Khalkhali|Marcolli|2008|p=171}} Perhaps one of the typical examples of a noncommutative space is the "[[noncommutative torus]]", which played a key role in the early development of this field in 1980s and lead to noncommutative versions of [[Vector bundle|vector bundles]], [[Connection (vector bundle)|connections]], [[curvature]], etc.{{sfn|Khalkhali|Marcolli|2008|p=21}}