In mathematics, a crossed product is a basic method of constructing a new von Neumann algebra from a von Neumann algebra acted on by a group.
This article concentrates on an important case, where they appear in functional analysis.)
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
We can generalize this construction further by replacing C[N] 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.
In physics, this structure appears in presence of the so called gauge group of the first kind.
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.
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.
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.
Moreover: In particular one can construct examples of all the different types of factors as crossed products.
is a von Neumann algebra on which a locally compact Abelian
: These unitaries normalise the crossed product, defining the dual action of
, which can be identified with the iterated crossed product by the dual action
: The crossed product may be identified with the fixed point algebra of the double dual action.
is replaced by a non-Abelian locally compact group or more generally a locally compact quantum group, a class of Hopf algebra related to von Neumann algebras.
An analogous theory has also been developed for actions on C* algebras and their crossed products.
Duality first appeared for actions of the reals in the work of Connes and Takesaki on the classification of Type III factors.
According to Tomita–Takesaki theory, every vector which is cyclic for the factor and its commutant gives rise to a 1-parameter modular automorphism group.
von Neumann algebra and the corresponding dual action restricts to an ergodic action of the reals on its centre, an Abelian von Neumann algebra.
The Connes spectrum, a closed subgroup of the positive reals
From this classification and results in ergodic theory, it is known that every infinite-dimensional hyperfinite factor has the form