Dual action on a crossed product - Imported Knowlpedia Content

Definition

Let $(A,G,\alpha)$ be a $C^*$ -dynamical system with $G$ locally compact abelian. The dual action is the strongly continuous action

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

on the full crossed product determined, for $\chi\in\widehat G$ , by

\widehat\alpha_\chi(i_A(a))=i_A(a),\qquad \widehat\alpha_\chi(i_G(s))=\overline{\chi(s)}\,i_G(s).

Equivalently, on the dense convolution algebra $C_c(G,A)$ ,

(\widehat\alpha_\chi f)(s)=\overline{\chi(s)}f(s).

It records the group variable by multiplying each Fourier mode by its character. The action fixes the coefficient algebra pointwise and changes only the canonical group unitaries. The complex conjugate reflects the Fourier-transform convention chosen here.

Why the formula defines an action

Multiplying $i_G(s)$ by the scalar $\overline{\chi(s)}$ preserves the covariance relation

i_G(s)i_A(a)i_G(s)^*=i_A(\alpha_s(a)).

The universal property of the crossed product therefore produces a unique automorphism. Character multiplication gives $\widehat\alpha_{\chi\psi}= \widehat\alpha_\chi\widehat\alpha_\psi$ , and the crossed-product norm plus density of $C_c(G,A)$ gives strong continuity Williams, Chapter 7.

The same formula descends through the regular representation and defines a dual action on the reduced crossed product.

Standard cases and conventions

If $A=\mathbb C$ and the action is trivial, the construction is the action of $\widehat G$ on $C^*(G)$ that becomes translation after Fourier transform. If $G$ is discrete, elements of the algebraic core have the form $\sum_s a_su_s$ , and

\widehat\alpha_\chi\left(\sum_s a_su_s\right) =\sum_s\overline{\chi(s)}a_su_s.

Some authors omit the complex conjugate in the defining formula. That amounts to replacing $\chi$ by $\chi^{-1}$ ; statements of duality must use one convention consistently.

References

Dana P. Williams, Crossed Products of $C^*$ -Algebras, Mathematical Surveys and Monographs 134, American Mathematical Society, 2007. AMS DOI record. Relevant: Chapter 7 on dual actions and crossed-product duality.
Gert K. Pedersen, $C^*$ -Algebras and Their Automorphism Groups, 2nd ed., Academic Press, 2018. Publisher DOI record. Relevant: Chapter 7 on crossed products and dual actions.