Noncommutative geometry

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Noncommutative geometry, or NCG, is a branch of mathematics concerned with the possible spatial interpretations of algebraic structures for which the commutative law fails, that is, for which xy does not always equal yx. For example; 3 steps of 4 units and 4 steps of 3 units length might be different in noncommutative spaces. Although one could technically construct geometries by simply removing this condition (commutativity), the results are typically trivial or uninteresting. The most common usage of the term, therefore, refers to what is properly called differential noncommutative geometry, a subject which was developed extensively by French mathematician Alain Connes. The challenge of NCG theory is to get around the lack of commutative multiplication, which is a requirement of previous geometric theories of algebraic structures.

1 Motivation
2 Non-commutative C*-algebras, von Neumann algebras
3 Non-commutative differentiable manifolds
4 Non-commutative affine schemes
5 Examples of non-commutative spaces
6 History
7 See also
8 Notes
9 External links

Motivation

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. For instance, one may take the ring C(X) of continuous complex-valued functions on a topological space X. In many important 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 geometry.

For other cases and applications, including mathematical physics^[1] and functional analysis, non-commutative rings arise as the natural candidates for a ring of functions on some non-commutative "space". "Non-commutative spaces", however defined, cannot be too similar to ordinary topological spaces, as these are known to correspond to commutative rings in many important cases. For this reason, the field is also called non-commutative topology — some of the motivating questions of the theory are concerned with extending known topological invariants to these new spaces.

Non-commutative C*-algebras, von Neumann algebras

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 S a topological space Ŝ; see spectrum of a C*-algebra.

For the duality between σ-finite measure spaces and commutative von Neumann algebras, noncommutative von Neumann algebras are called non-commutative measure spaces.

Non-commutative differentiable manifolds

A smooth Riemannian manifold M is a topological space with a lot of extra structure. From its algebra of continuous functions C(M) we only recover M topologically. The algebraic invariant that recovers the Riemannian structure is a spectral triple. It is constructed from a smooth vector bundle E over M, e.g. the exterior algebra bundle. The Hilbert space L²(M,E) of square integrable sections of E carries a representation of C(M) by multiplication operators, and we consider an unbounded operator D in L²(M,E) with compact resolvent (e.g the signature operator), such that the commutators [D,f] are bounded whenever f is smooth. A recent deep theorem states that M as a Riemannian manifold can be recovered from this data.

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. Research in spectral triples is very active, and many examples of noncommutative manifolds have been constructed.

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.

Non-commutative affine schemes

In analogy to the duality between affine schemes and commutative rings, we can also have noncommutative affine schemes. For example, there exist an analog of the celebrated Serre duality for noncommutative projective schemes. [1]

Examples of non-commutative spaces

In Weyl quantization, the symplectic phase space of classical mechanics is deformed into a non-commutative phase space generated by the position and momentum operators.

The standard model of particle physics is another example of a noncommutative geometry, cf noncommutative standard model.

The noncommutative torus, deformation of the function algebra of the ordinary torus, can be given the structure of a spectral triple. This class of examples has been studied intensively and still functions as a test case for more complicated situations.

Nonncommutative algebras arising from foliations.

Examples related to dynamical systems arising from number theory, such as the Gauss shift on continued fractions, give rise to noncommutative algebras that appear to have interesting noncommutative geometries.

History

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.