---
tag: PS2MR7P299
kind: concept
name: Noncommutative geometry
msc: [58B34, 46L87, 46L85]
sources:
  - url: https://en.wikipedia.org/wiki/Noncommutative_geometry
same_as:
  wikidata: Q2746792
  wikipedia: Noncommutative geometry
---

## Summary

Noncommutative geometry studies algebras that are not commutative as if they were algebras of functions on a space.
Its starting point is the [[commutative Gelfand–Naimark theorem|NT564Y315M]].
By this theorem, a locally compact Hausdorff space is determined by its C*-algebra of continuous functions that vanish at infinity.
A noncommutative C*-algebra is then read as a "noncommutative space", and tools of topology and geometry are carried over to it.
The theory gives a geometric language for spaces of orbits, foliations, quantum physics and number theory, where the usual point sets behave badly.

## Explanations

### L1 human

On an ordinary space, functions can be multiplied in any order: $fg = gf$.
Much of geometry can be recovered from these functions alone, without looking at the points.
Noncommutative geometry keeps the algebra and drops the rule $fg = gf$.
A standard example is the [[noncommutative torus|PZ97S3QQF1]], made of two unitaries $u$ and $v$ with $vu = e^{2\pi i \theta} uv$.
For $\theta = 0$ it is the algebra of continuous functions on the ordinary torus, and for other $\theta$ it behaves like a "quantum" torus.
Geometric ideas such as K-theory, distance and dimension still make sense for such algebras.