Crossed product

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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.)

1 Motivation
2 Construction
3 Properties
4 Examples
5 References

[edit] Motivation

Recall that if we have two finite groups G and N with an action of G on N we can form the semidirect product NG. 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 C[N]G 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 NG. We can generalize this construction further by replacing C[N] by any algebra A acted on by G to get a crossed product AG, 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 then the algebraic crossed product discussed above; in fact it is some sort of completion of the algebraic crossed product.)

[edit] Construction

Suppose that A is an abelian 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 AG is the von Neumann algebra acting on K generated by the actions of A and G on H. 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.

[edit] 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 M acted on by G, and then the action of G on M 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 AG 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 M being transitive. Example: M 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 M having a finite G invariant measure, absolutely continuous with respect to the measure on M. Example: M 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 M having an infinite G invariant measure without atoms, absolutely continuous with respect to the measure on M. Example: M 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 M having no non-zero absolutely continuous G-invariant measure. Example: M 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.

[edit] Examples

If we take the algebra A to be the complex numbers C, then the crossed product AG is called the von Neumann group algebra of G.

If G is an infinite discrete group such that every conjugacy class has infinite order then the von Neumann group algebra is a factor of type II₁. Moreover if every finite set of elements of G generates a finite subgroup (or more generally if G is amenable) then the factor is the hyperfinite factor of type II₁.