Dual action on a von Neumann crossed product - Imported Knowlpedia Content

Definition

Let an abelian locally compact group $G$ act point-ultraweakly continuously on a von Neumann algebra $M$ by $\alpha$ . The dual action on the von Neumann crossed product is the point-ultraweakly continuous action of the Pontryagin dual $\widehat G$

\widehat\alpha:\widehat G\longrightarrow \operatorname{Aut}(M\rtimes_\alpha G)

determined by

\widehat\alpha_\chi(\pi_\alpha(x))=\pi_\alpha(x),\qquad \widehat\alpha_\chi(\lambda(t)) =\overline{\chi(t)}\,\lambda(t).

Thus it fixes the coefficient algebra pointwise and multiplies the canonical group unitary of $t$ by its Fourier character.

Existence and uniqueness

The displayed assignments preserve the covariance relation and therefore extend from integrated covariant operators to normal automorphisms of the generated von Neumann algebra. Character multiplication gives an action, and ultraweak density of the integrated core gives uniqueness. When $G=\mathbb R$ and $\chi_s(t)=e^{ist}$ , the convention above reads $\widehat\alpha_s(\lambda(t))=e^{-ist}\lambda(t)$ .

Duality

Crossing again by $\widehat G$ recovers $M$ , stabilized by $\mathcal B(L^2(G))$ , through Takesaki duality. The corresponding double-dual action remembers the original action up to the standard inner correction. This makes the dual action structural data rather than an arbitrary symmetry of the crossed product Nakagami–Takesaki, Chapters 1–2.

Relation to other dual actions

The formula resembles the dual action on a $C^*$ -crossed product, but the topology and ambient algebra differ: point-ultraweak continuity and normal automorphisms are used here. For nonabelian $G$ , the replacement is generally a dual coaction rather than an action of a Pontryagin dual group.

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, coactions, crossed products, and duality.
Masamichi Takesaki, Theory of Operator Algebras II, Springer, 2003. Publisher DOI record. Relevant: Chapter X on dual actions and Takesaki duality.