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.) Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... A von Neumann algebra is a *-algebra of bounded operators on a Hilbert space which is closed in the weak operator topology, or equivalently, in the strong operator topology (under pointwise convergence) and contains the identity operator. ... In abstract algebra, a semidirect product describes a particular way in which a group can be put together from two subgroups. ... In mathematics, the group ring is an algebraic construction that associates to a group G and a commutative ring with unity R an R-algebra R[G] (or sometimes just RG) such that the multiplication in R[G] is induced by the multiplication in G. R[G] can be described... In mathematics, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ...

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 N G. 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 N G. We can generalize this construction further by replacing C[N] by any algebra A acted on by G to get a crossed product A G, which is the sum of subspaces gA and where the action of G on A is given by conjugation in the crossed product. This is the symbol for the semidirect product operation, until Wikipedia:TeX support for semidirect is implemented (rtimes is the standard LaTeX command, actually) This image has been released into the public domain by the copyright holder, its copyright has expired, or it is ineligible for copyright. ... This is the symbol for the semidirect product operation, until Wikipedia:TeX support for semidirect is implemented (rtimes is the standard LaTeX command, actually) This image has been released into the public domain by the copyright holder, its copyright has expired, or it is ineligible for copyright. ... This is the symbol for the semidirect product operation, until Wikipedia:TeX support for semidirect is implemented (rtimes is the standard LaTeX command, actually) This image has been released into the public domain by the copyright holder, its copyright has expired, or it is ineligible for copyright. ... This is the symbol for the semidirect product operation, until Wikipedia:TeX support for semidirect is implemented (rtimes is the standard LaTeX command, actually) This image has been released into the public domain by the copyright holder, its copyright has expired, or it is ineligible for copyright. ...

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.)

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 In functional analysis, an Abelian von Neumann algebra is a von Neumann algebra of operators on a Hilbert space in which all its elements commute. ...

• 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 A G 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 is the symbol for the semidirect product operation, until Wikipedia:TeX support for semidirect is implemented (rtimes is the standard LaTeX command, actually) This image has been released into the public domain by the copyright holder, its copyright has expired, or it is ineligible for copyright. ...

This construction can be extended to work for any locally compact group G acting on any von Neumann algebra A.

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. In mathematics, a measure is a function that assigns a number, e. ...

If the action of G on A is free and ergodic then the crossed product A G is a factor. Moreover: This is the symbol for the semidirect product operation, until Wikipedia:TeX support for semidirect is implemented (rtimes is the standard LaTeX command, actually) This image has been released into the public domain by the copyright holder, its copyright has expired, or it is ineligible for copyright. ...

• 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.

Examples

• If we take the algebra A to be the complex numbers C, then the crossed product A G 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.

This is the symbol for the semidirect product operation, until Wikipedia:TeX support for semidirect is implemented (rtimes is the standard LaTeX command, actually) This image has been released into the public domain by the copyright holder, its copyright has expired, or it is ineligible for copyright. ...

References

• Theory of Operator Algebras I, II, III by M. Takesaki, ISBN 3-540-42248-X ISBN 3-540-42914-X ISBN 3-540-42913-1
• Non-commutative geometry by A. Connes, ISBN 0-12-185860-X