Von Neumann crossed product

Let $M\subseteq\mathcal B(H)$ be a von Neumann algebra, let $G$ be a locally compact group, and let $\alpha:G\to\operatorname{Aut}(M)$ be a point-ultraweakly continuous action. On $L^2(G,H)$ , set

(\pi_\alpha(x)\xi)(s)=\alpha_{s^{-1}}(x)\xi(s),\qquad (\lambda(t)\xi)(s)=\xi(t^{-1}s).

The von Neumann crossed product is

M\rtimes_\alpha G= \bigl(\pi_\alpha(M)\cup\lambda(G)\bigr)'',

the von Neumann algebra generated by this covariant pair. Its isomorphism class is independent of the faithful normal realization of $M$ .

Covariance and generators

The defining operators satisfy

\lambda(t)\pi_\alpha(x)\lambda(t)^* =\pi_\alpha(\alpha_t(x)).

Thus the crossed product contains a normal copy of $M$ and unitaries implementing the action. For a discrete group, finite sums $\sum_t x_t\lambda(t)$ form an ultraweakly dense algebraic core. For a general locally compact group, integrated operators $\int_G\pi_\alpha(f(t))\lambda(t)\,dt$ provide the corresponding core.

Standard examples

For the trivial action on $\mathbb C$ , the crossed product is the group von Neumann algebra $L(G)$ . For a measure-preserving action on a standard measure space, the crossed product $L^\infty(X,\mu)\rtimes G$ is the group-measure-space von Neumann algebra. Inner actions often yield tensor-product models, whereas outer actions can change the factor type.

Distinction from C*-crossed products

This construction takes a bicommutant, or equivalently an ultraweak closure, in a normal covariant representation. It is therefore not interchangeable with either the full or reduced $C^*$ -crossed product, which are defined for point-norm-continuous $C^*$ -dynamical systems. When the action on $M$ , regarded as a $C^*$ -algebra, is also point-norm continuous, its regular covariant representation gives a represented reduced $C^*$ -crossed product; the ultraweak closure of that represented algebra is the von Neumann crossed product above. For a merely point-ultraweakly continuous action, this comparison does not assert the existence of a reduced $C^*$ -crossed product for the same action.

References

Yoshiomi Nakagami and Masamichi Takesaki, Duality for Crossed Products of von Neumann Algebras, Lecture Notes in Mathematics 731, Springer, 1979. Publisher DOI record. Relevant: Chapters 1–2 on actions, covariant representations, and elementary properties of crossed products.
Masamichi Takesaki, Theory of Operator Algebras II, Springer, 2003. Publisher DOI record. Relevant: Chapter X on crossed products of von Neumann algebras.