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. 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.