Noncommutative geometry

Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of spaces that are locally presented by noncommutative algebras of functions (possibly in some generalized sense). A noncommutative algebra is an associative algebra in which the multiplication is not commutative, that is, for which does not always equal ; or more generally an algebraic structure in which one of the principal binary operations is not commutative; one also allows additional structures, e.g. topology or norm, to be possibly carried by the noncommutative algebra of functions. An approach giving deep insight about noncommutative spaces is through operator algebras (i.e. algebras of bounded linear operators on a Hilbert space). Perhaps one of the typical examples of a noncommutative space is the "noncommutative tori", which played a key role in the early development of this field in 1980s and lead to noncommutative versions of vector bundles, connections, curvature, etc. The main motivation is to extend the commutative duality between spaces and functions to the noncommutative setting. In mathematics, spaces, which are geometric in nature, can be related to numerical functions on them. In general, such functions will form a commutative ring. For instance, one may take the ring C(X) of continuous complex-valued functions on a topological space X. 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. More specifically, in topology, compact Hausdorff topological spaces can be reconstructed from the Banach algebra of functions on the space (Gelfand–Naimark). In commutative algebraic geometry, algebraic schemes are locally prime spectra of commutative unital rings (A. Grothendieck), and every quasi-separated scheme can be reconstructed up to isomorphism of schemes from the category of quasicoherent sheaves of -modules (P. Gabriel–A. Rosenberg).