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.)
Motivation
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.
Construction
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. We let K be the Hilbert space of all square summable H-valued functions on G. There is an action of A on K given by a(k)(g) = g−1(a)k(g) for k in K, g, h in G, and a in A, and there is an action of G on K given by g(k)(h) = k(g−1h). The crossed product is the von Neumann algebra acting on K generated by the actions of A and G on K. It does not depend (up to isomorphism) on the choice of the Hilbert space H.
This construction can be extended to work for any locally compact group G acting on any von Neumann algebra A. When is an abelian von Neumann algebra, this is the original group-measure space construction of Murray and von Neumann.
Properties
We let G be an infinite countable discrete group acting on the abelian von Neumann algebra A. The action is called free if A has no non-zero projections p such that some nontrivial g fixes all elements of pAp. The action is called ergodic if the only invariant projections are 0 and 1. Usually A can be identified as the abelian von Neumann algebra of essentially bounded functions on a measure space X acted on by G, and then the action of G on X is ergodic (for any measurable invariant subset, either the subset or its complement has measure 0) if and only if the action of G on A is ergodic.
If the action of G on A is free and ergodic
then the crossed product is a factor. Moreover: The factor is of type I if A has a minimal projection such that 1 is the sum of the G conjugates of this projection. This corresponds to the action of G on X being transitive. Example: X is the integers, and G is the group of integers acting by translations. The factor has type II1 if A has a faithful finite normal G-invariant trace. This corresponds to X having a finite G invariant measure, absolutely continuous with respect to the measure on X. Example: X is the unit circle in the complex plane, and G is the group of all roots of unity. The factor has type II∞ if it is not of types I or II1 and has a faithful semifinite normal G-invariant trace. This corresponds to X having an infinite G invariant measure without atoms, absolutely continuous with respect to the measure on X. Example: X is the real line, and G is the group of rationals acting by translations. The factor has type III if A has no faithful semifinite normal G-invariant trace. This corresponds to X having no non-zero absolutely continuous G-invariant measure. Example: X is the real line, and G is the group of all transformations ax+b for a and b rational, a non-zero.
In particular one can construct examples of all the different types of factors as crossed products.