Takai duality - Imported Knowlpedia Content

Statement

Let $(A,G,\alpha)$ be a $C^*$ -dynamical system with $G$ locally compact abelian, and let $\widehat\alpha$ be the dual action on $A\rtimes_\alpha G$ . Takai duality gives a canonical $*$ -isomorphism

(A\rtimes_\alpha G)\rtimes_{\widehat\alpha}\widehat G \cong A\otimes K(L^2(G)).

Here $K(L^2(G))$ is the compact-operator $C^*$ -algebra. Thus crossing twice recovers $A$ after stabilization; it does not generally recover $A$ literally. The theorem is natural for equivariant $*$ -homomorphisms and also identifies the double-dual action, so it retains dynamical information rather than giving only an abstract algebra isomorphism.

Equivariance

Pontryagin duality identifies $\widehat{\widehat G}$ with $G$ , so the double crossed product carries a double-dual action. Under a standard Takai isomorphism this action corresponds to

s\longmapsto \alpha_s\otimes\operatorname{Ad}\rho_s,

where $\rho$ is a regular representation of $G$ on $L^2(G)$ . Left-versus-right regular-representation and character conventions can change the displayed implementation by inversion or unitary conjugacy, but not the stabilized isomorphism class Takai, pp. 25–39.

How to interpret the theorem

The first crossed product packages the coefficients $A$ together with the $G$ -action. The dual action then measures the Fourier variable introduced by that construction. Crossing by $\widehat G$ performs the inverse Fourier operation, while the regular representation leaves the compact operator factor.

Because $K(L^2(G))$ is Morita equivalent to $\mathbb C$ , Takai duality implies that the double crossed product is strongly Morita equivalent to $A$ . Consequently stable invariants, including $K$ -theory, return to those of the coefficient algebra.

Full and reduced versions

For abelian $G$ , the group is amenable. Hence full and reduced crossed products agree at both stages, so the theorem may be written with either completion. For nonabelian groups, crossed-product duality is formulated using coactions rather than a Pontryagin-dual group action.

References

Hiroshi Takai, “On a duality for crossed products of $C^*$ -algebras,” Journal of Functional Analysis 19 (1975), 25–39. Publisher DOI record. Relevant: the original stabilized double-crossed-product theorem.
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 Takai duality.