In mathematics, there is a close relationship between spaces, which are geometric in nature, and the numerical functions on them. In general such functions will form a commutative ring, say the ring C(X) of continuous functions on a topological space X, with complex values. In many important cases (X a compact Hausdorff space), we can recover X from C(X); it makes some sense to say that X has commutative geometry.
For other cases and applications, including in relation with mathematical physics1, and in particular in functional analysis, non-commutative rings arise. In a general sense non-commutative geometry investigates the possible spatial interpretations of such rings; the point is to get round the lack of commutative multiplication, which is a requirement of any simple-minded spatial theory. Non-commutative spaces, when introduced, cannot be too similar to ordinary spaces. The field is also called non-commutative topology — some of the motivating examples are questions of extending known topological invariants to such . That is, the 'space' itself is used as some sort of middle term
Non-commutative C* algebras are often now called non-commutative spaces. This is by analogy with the Gelfand representation, which shows that commutative C*-algebras are dual to locally compact Hausdorff spaces. In general, one can associate to any C*-algebra A a topological space Â; see spectrum of a C*-algebra.
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 actions would become homogeneous spaces of an extended kind, has by now been subsumed.
How about noncommutative differential manifolds? Well, first, for an ordinary differential manifold, we can look at the commutative algebra of smooth functions over it and also the space of smooth sections of its tangent bundle, cotangent bundle and other fiber bundles. All these spaces are modules of the commutative algebra. We'd also need the concept of exterior derivative and/or Lie derivative and/or covariant derivative. We note that they act as derivations over the algebra in question. So, to go over to the noncommutative case, we simply make the algebras in question noncommutative. To handle differential forms, we work with the graded exterior algebra bundle of all p-forms under the wedge product and look at its algebra of smooth sections. A differential is an antiderivation (or something more general) on this algebra which increases the grading by 1 and is quadratically nilpotent.
For the duality between locally compact measure spaces and commutative von Neumann algebras, we could call noncommutative von Neumann algebras non-commutative measure spaces.
Examples
Also, in analogy to the duality between affine schemes and polynomial algebras, we can also have noncommutative affine schemes.
[1] The applications in particle physics are described on the entry for Noncommutative quantum field theory