Crossed Product - Construction

Construction

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 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 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 is an abelian von Neumann algebra, this is the original group-measure space construction of Murray and von Neumann.