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  "body": "\n## Summary\n\nNoncommutative geometry studies algebras that are not commutative as if they were algebras of functions on a space.\nIts starting point is the [[commutative Gelfand–Naimark theorem|NT564Y315M]].\nBy this theorem, a locally compact Hausdorff space is determined by its C*-algebra of continuous functions that vanish at infinity.\nA noncommutative C*-algebra is then read as a \"noncommutative space\", and tools of topology and geometry are carried over to it.\nThe theory gives a geometric language for spaces of orbits, foliations, quantum physics and number theory, where the usual point sets behave badly.\n\n## Explanations\n\n### L1 human\n\nOn an ordinary space, functions can be multiplied in any order: $fg = gf$.\nMuch of geometry can be recovered from these functions alone, without looking at the points.\nNoncommutative geometry keeps the algebra and drops the rule $fg = gf$.\nA standard example is the [[noncommutative torus|PZ97S3QQF1]], made of two unitaries $u$ and $v$ with $vu = e^{2\\pi i \\theta} uv$.\nFor $\\theta = 0$ it is the algebra of continuous functions on the ordinary torus, and for other $\\theta$ it behaves like a \"quantum\" torus.\nGeometric ideas such as K-theory, distance and dimension still make sense for such algebras.\n",
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