Crossed product

In mathematics, and more specifically in the theory of von Neumann algebras, a crossed productis 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.) In mathematics, and more specifically in the theory of von Neumann algebras, a crossed productis 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 G {displaystyle G} and N with an action of G on N we can form the semidirect product N ⋊ G {displaystyle N times G} . This contains Nas 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, and again form a product C [ N ] ⋊ G {displaystyle C times G} in a similar way; this algebra is a sum of subspaces gC as g runs through the elements of G, and is the group algebra of N ⋊ G {displaystyle N times G} .We can generalize this construction further by replacing Cby any algebra A acted on by G to get a crossed product A ⋊ G {displaystyle A times G} , which is the sum of subspacesgA 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. We let K be the Hilbert space of all square summable H-valued functions on G. There is an action of A on Kgiven by for k in K, g, h in G, and a in A,and there is an action of G on K given by The crossed product A ⋊ G {displaystyle A times G} 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 A {displaystyle A} is an abelian von Neumann algebra, this is the original group-measure space construction of Murray and von Neumann. We let G be an infinite countable discrete group acting on the abelian von Neumann algebra A. The action is called free ifA 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 L ∞ ( X ) {displaystyle L^{infty }(X)} 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.