Green imprimitivity theorem - Imported Knowlpedia Content

Statement

Let $G$ be a locally compact group, $H\subseteq G$ a closed subgroup, and $(A,H,\alpha)$ a $C^*$ -dynamical system. Form the induced algebra

\operatorname{Ind}_H^G(A,\alpha) =\{f\in C_b(G,A): f(sh)=\alpha_{h^{-1}}(f(s)),\quad sH\mapsto\lVert f(s)\rVert\in C_0(G/H)\},

with $G$ acting by left translation. The Green imprimitivity theorem states that the full crossed products

\operatorname{Ind}_H^G(A,\alpha)\rtimes G \quad\text{and}\quad A\rtimes_\alpha H

are strongly Morita equivalent through a canonical imprimitivity bimodule obtained by completing $C_c(G,A)$ .

Representation-theoretic content

The Green bimodule implements Rieffel induction between the nondegenerate representation categories of the two crossed products. Under the crossed-product/covariant-representation correspondence, this recovers induction from $H$ -covariant representations to $G$ -covariant representations. The precise construction and inner products are developed in Williams, Chapter 4.

Homogeneous-space form

If $\alpha$ is the restriction to $H$ of an action of $G$ on $A$ , the induced algebra can be identified with a diagonal-action model based on $C_0(G/H,A)$ . In particular, for $A=\mathbb C$ with the trivial action, the theorem says

C_0(G/H)\rtimes G\ \sim_M\ C^*(H).

This is the crossed-product form of Mackey's imprimitivity theorem.

Conventions and scope

The displayed theorem concerns full crossed products and an arbitrary closed subgroup. Reduced crossed-product analogues require their own formulation and hypotheses and should not be inferred by simply adding subscripts $r$ . Different sources use right-coset conventions or write the covariance condition with $\alpha_h$ rather than $\alpha_{h^{-1}}$ ; changing both the translation and covariance conventions yields an equivalent induced system.

References

Philip Green, “The Local Structure of Twisted Covariance Algebras,” Acta Mathematica 140 (1978), 191–250. DOI record. Relevant: the imprimitivity theorem for induced covariance algebras.
Dana P. Williams, Crossed Products of $C^*$ -Algebras, Mathematical Surveys and Monographs 134, American Mathematical Society, 2007. DOI record. Relevant: Chapter 4 on induced algebras and Green's imprimitivity theorem.