Crossed Product - Properties

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.