Editing Noncommutative geometry (section)

==Motivation==

The main motivation is to extend the commutative duality between spaces and functions to the noncommutative setting. In mathematics, ''[[Space (mathematics)|spaces]]'', which are geometric in nature, can be related to numerical [[function (mathematics)|functions]] on them. In general, such functions will form a [[commutative ring]]. For instance, one may take the ring ''C''(''X'') of [[continuous function|continuous]] [[complex number|complex]]-valued functions on a [[topological space]] ''X''. In many cases (''e.g.'', if ''X'' is a [[Compact space|compact]] [[Hausdorff space]]), we can recover ''X'' from ''C''(''X''), and therefore it makes some sense to say that ''X'' has ''commutative topology''.

More specifically, in topology, compact [[Hausdorff space|Hausdorff]] topological spaces can be reconstructed from the [[Banach algebra]] of functions on the space ([[Gelfand representation#Statement of the commutative Gelfand–Naimark theorem|Gelfand–Naimark]]). In commutative [[algebraic geometry]], [[Scheme (algebraic geometry)|algebraic scheme]]s are locally prime spectra of commutative unital rings ([[Alexander Grothendieck|A. Grothendieck]]), and every quasi-separated scheme <math>X</math> can be reconstructed up to isomorphism of schemes from the category of quasicoherent sheaves of <math>O_X</math>-modules ([[Pierre Gabriel|P. Gabriel]]–A. Rosenberg). For [[Grothendieck topologies]], the cohomological properties of a site are invariants of the corresponding category of sheaves of sets viewed abstractly as a [[topos]] (A. Grothendieck). In all these cases, a space is reconstructed from the algebra of functions or its categorified version—some [[Sheaf (mathematics)|category of sheaves]] on that space.

Functions on a topological space can be multiplied and added pointwise hence they form a commutative algebra; in fact these operations are local in the topology of the base space, hence the functions form a sheaf of commutative rings over the base space.

The dream of noncommutative geometry is to generalize this duality to the duality between noncommutative algebras, or sheaves of noncommutative algebras, or sheaf-like noncommutative algebraic or operator-algebraic structures, and geometric entities of certain kinds, and give an interaction between the algebraic and geometric description of those via this duality.

Regarding that the commutative rings correspond to usual affine schemes, and commutative [[C*-algebras]] to usual topological spaces, the extension to noncommutative rings and algebras requires non-trivial generalization of [[topological space]]s as "non-commutative spaces". For this reason there is some talk about [[non-commutative topology]], though the term also has other meanings.

===Applications in mathematical physics===

There is an influence of physics on noncommutative geometry.<ref>{{cite journal | last1=Connes | first1=Alain | last2=Douglas | first2=Michael R | last3=Schwarz | first3=Albert | title=Noncommutative geometry and Matrix theory | journal=[[Journal of High Energy Physics]] | volume=1998 | issue=2 | date=1998-02-05 | issn=1029-8479 | doi=10.1088/1126-6708/1998/02/003 | pages=003|arxiv=hep-th/9711162| bibcode=1998JHEP...02..003C | s2cid=7562354 }}</ref> The [[fuzzy sphere]] has been used to study the emergence of [[conformal symmetry]] in the 3-dimensional [[Ising model]].<ref>{{Cite journal |last=Zhu |first=Wei |last2=Han |first2=Chao |last3=Huffman |first3=Emilie |last4=Hofmann |first4=Johannes S. |last5=He |first5=Yin-Chen |date=2023-04-18 |title=Uncovering Conformal Symmetry in the 3D Ising Transition: State-Operator Correspondence from a Quantum Fuzzy Sphere Regularization |url=https://journals.aps.org/prx/abstract/10.1103/PhysRevX.13.021009 |journal=Physical Review X |volume=13 |issue=2 |pages=021009 |arxiv=2210.13482 |doi=10.1103/PhysRevX.13.021009}}</ref>

===Motivation from ergodic theory===
Some of the theory developed by [[Alain Connes]] to handle noncommutative geometry at a technical level has roots in older attempts, in particular in [[ergodic theory]]. The proposal of [[George Mackey]] to create a ''virtual subgroup'' theory, with respect to which ergodic [[Group action (mathematics)|group action]]s would become [[homogeneous space]]s of an extended kind, has by now been subsumed.