Crossed Product - Motivation

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, and again form a product 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 . We can generalize this construction further by replacing C 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.)