[2] The main motivation is to extend the commutative duality between spaces and functions to the noncommutative setting.
In many cases (e.g., if X is a compact Hausdorff space), we can recover X from C(X), and therefore it makes some sense to say that X has commutative topology.
The sudden rise in interest in noncommutative geometry in physics follows after the speculations of its role in M-theory made in 1997.
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.
This is by analogy with the Gelfand representation, which shows that commutative C*-algebras are dual to locally compact Hausdorff spaces.
This suggests that one might define a noncommutative Riemannian manifold as a spectral triple (A, H, D), consisting of a representation of a C*-algebra A on a Hilbert space H, together with an unbounded operator D on H, with compact resolvent, such that [D, a] is bounded for all a in some dense subalgebra of A.
There are certain analogues of Zariski topology in that context so that one can glue such affine schemes to more general objects.
This theorem is extended as a definition of noncommutative projective geometry by Michael Artin and J. J. Zhang,[5] who add also some general ring-theoretic conditions (e.g. Artin–Schelter regularity).
For example, there exists an analog of the celebrated Serre duality for noncommutative projective schemes of Artin and Zhang.
[6] A. L. Rosenberg has created a rather general relative concept of noncommutative quasicompact scheme (over a base category), abstracting Grothendieck's study of morphisms of schemes and covers in terms of categories of quasicoherent sheaves and flat localization functors.
[7] There is also another interesting approach via localization theory, due to Fred Van Oystaeyen, Luc Willaert and Alain Verschoren, where the main concept is that of a schematic algebra.
The theory of characteristic classes of smooth manifolds has been extended to spectral triples, employing the tools of operator K-theory and cyclic cohomology.
Several generalizations of now-classical index theorems allow for effective extraction of numerical invariants from spectral triples.
The fundamental characteristic class in cyclic cohomology, the JLO cocycle, generalizes the classical Chern character.