Produit croisé

In mathematics, and more specifically in the theory of von Neumann algebras, a crossed product is a basic method of constructing a new von Neumann algebra from a von Neumann algebra acted on by a group. It is related to the semidirect product construction for groups. (Roughly speaking, crossed product is the expected structure for a group ring of a semidirect product group. Therefore crossed products have a ring theory aspect also. This article concentrates on an important case, where they appear in functional analysis.) Recall that if we have two finite groups and N with an action of G on N we can form the semidirect product . This contains N as a normal subgroup, and the action of G on N is given by conjugation in the semidirect product. We can replace N by its complex group algebra C[N], and again form a product in a similar way; this algebra is a sum of subspaces gC[N] as g runs through the elements of G, and is the group algebra of . We can generalize this construction further by replacing C[N] by any algebra A acted on by G to get a crossed product which is the sum of subspaces gA and where the action of G on A is given by conjugation in the crossed product. The crossed product of a von Neumann algebra by a group G acting on it is similar except that we have to be more careful about topologies, and need to construct a Hilbert space acted on by the crossed product. (Note that the von Neumann algebra crossed product is usually larger than the algebraic crossed product discussed above; in fact it is some sort of completion of the algebraic crossed product.) In physics, this structure appears in presence of the so called gauge group of the first kind. G is the gauge group, and N the "field" algebra. The observables are then defined as the fixed points of N under the action of G. A result by Doplicher, Haag and Roberts says that under some assumptions the crossed product can be recovered from the algebra of observables. Suppose that A is a von Neumann algebra of operators acting on a Hilbert space H and G is a discrete group acting on A.