Induced \(C^*\)-algebras, coactions and equivariance in the symmetric imprimitivity theorem.

*(English)*Zbl 0982.46053In a similar way that G acts as a transformation group on a locally compact space it can act as a group of automorphisms on a C\(^{*}\)-algebra \(D\) and one may use a non-commutative dynamical system consisting of a crossed product \( D \times_{\alpha} G\) for an action \(\alpha\) of \(G\) as \(D\). For a closed subgroup \(K\) of a locally compact group \(G\), G. Mackey’s imprimitivity theorem gives a Morita equivalence of the group C\(^{*}\)-algebra \(C^{*}(K)\) and the transformation group C\(^{*}\)-algebra \(C^{*}(G,G/K)\); the symmetric version, as in the title, includes another closed subgroup \(H\) which acts on the left of the right coset space \(K \backslash G\), giving a Morita equivalence of \(C^{*}(H,K \backslash G)\) and \(C^{*}(K,G/H)\) [M. A. Rieffel, Math. Ann. 222, 7-22 (1976; Zbl 0328.22013)].

Suppose \(\alpha\) and \( \beta\) commuting actions of K and H, respectively, as automorphisms of \(D\). Denote by \(\tau\) (or \(\sigma\)) the action of \(K\) (or \(H\)) by left (or right) translation. Abbreviating the notation in the paper, \(I^{G}_{H,\beta}\) denotes a C\(^{*}\)-algebra, of continuous \(D\)-valued functions on G, induced from the action \(\beta\) of \(H\). One constructs the dynamical system \(I^{G}_{H,\beta} \times K\) where the action of \(K\) is by \(\tau \otimes \alpha\); this denoting the restriction to \(I^{G}_{H,\beta}\) of the “diagonal action” C\(^{}\)-algebra on the above C\(^{*}\)-algebra of continuous \(D\)-valued functions. The same procedure is used to construct \(I^{G}_{K,\alpha} \times H\) with the action \(\sigma \otimes \beta\).

I. Raeburn [Math. Ann. 280, No. 3, 369-387 (1988; Zbl 0617.22009)] describes the Morita equivalence between these two crossed products, constructing the bimodule implementing the equivalence. The authors look to whether the symmetric imprimitivity theorem of Raeburn (loc. cit.) has an equivariant version in that the bimodules carry actions, or co-actions, compatible with those on the crossed products.

For locally compact Abelian \(G\) the authors reduce the result to actions of the dual \({\widehat{G}}\) in that bimodules implementing Morita equivalence carry actions of \({\widehat{G}}\) compatible with those on the crossed products. In order to do this the authors “inflate” the diagonal actions to actions of \({\widehat{G}}\) by composing the dual actions \({(\tau \otimes \alpha)}\widehat{\;}\) and \({(\sigma \otimes \beta)} \widehat{\;}\) with the restriction mappings to \(K\) and \(H\) respectively.

A co-action of \(G\) on a C\(^{*}\)-algebra \(A\) [cf. M. B. Landstad, Trans. Am. Math. Soc. 248, 223-267 (1979; Zbl 0397.46059)]is generally a 1-1 *-homomorphism \(A \to M(A \otimes C^{*}_{r}(G)\), where \( C^{*}_{r}(G)\) denotes the reduced group algebra and \(M\) indicates the multiplier algebra; for Abelian G a co-action is equivalent to the action of \({\widehat{G}}\) on A. The authors inflate co-actions of \(H\) and \(K\) to co-actions of \(G\). They restrict the dynamical systems to reduced \(C^{*}\)-algebras and reduced crossed products to be able to show that Raeburn’s symmetric imprimitivity theorem for non-commutative systems holds for these inflated co-actions.

Suppose \(\alpha\) and \( \beta\) commuting actions of K and H, respectively, as automorphisms of \(D\). Denote by \(\tau\) (or \(\sigma\)) the action of \(K\) (or \(H\)) by left (or right) translation. Abbreviating the notation in the paper, \(I^{G}_{H,\beta}\) denotes a C\(^{*}\)-algebra, of continuous \(D\)-valued functions on G, induced from the action \(\beta\) of \(H\). One constructs the dynamical system \(I^{G}_{H,\beta} \times K\) where the action of \(K\) is by \(\tau \otimes \alpha\); this denoting the restriction to \(I^{G}_{H,\beta}\) of the “diagonal action” C\(^{}\)-algebra on the above C\(^{*}\)-algebra of continuous \(D\)-valued functions. The same procedure is used to construct \(I^{G}_{K,\alpha} \times H\) with the action \(\sigma \otimes \beta\).

I. Raeburn [Math. Ann. 280, No. 3, 369-387 (1988; Zbl 0617.22009)] describes the Morita equivalence between these two crossed products, constructing the bimodule implementing the equivalence. The authors look to whether the symmetric imprimitivity theorem of Raeburn (loc. cit.) has an equivariant version in that the bimodules carry actions, or co-actions, compatible with those on the crossed products.

For locally compact Abelian \(G\) the authors reduce the result to actions of the dual \({\widehat{G}}\) in that bimodules implementing Morita equivalence carry actions of \({\widehat{G}}\) compatible with those on the crossed products. In order to do this the authors “inflate” the diagonal actions to actions of \({\widehat{G}}\) by composing the dual actions \({(\tau \otimes \alpha)}\widehat{\;}\) and \({(\sigma \otimes \beta)} \widehat{\;}\) with the restriction mappings to \(K\) and \(H\) respectively.

A co-action of \(G\) on a C\(^{*}\)-algebra \(A\) [cf. M. B. Landstad, Trans. Am. Math. Soc. 248, 223-267 (1979; Zbl 0397.46059)]is generally a 1-1 *-homomorphism \(A \to M(A \otimes C^{*}_{r}(G)\), where \( C^{*}_{r}(G)\) denotes the reduced group algebra and \(M\) indicates the multiplier algebra; for Abelian G a co-action is equivalent to the action of \({\widehat{G}}\) on A. The authors inflate co-actions of \(H\) and \(K\) to co-actions of \(G\). They restrict the dynamical systems to reduced \(C^{*}\)-algebras and reduced crossed products to be able to show that Raeburn’s symmetric imprimitivity theorem for non-commutative systems holds for these inflated co-actions.

Reviewer: I.Valuşescu (Bucureşti)

##### MSC:

46L55 | Noncommutative dynamical systems |

22D30 | Induced representations for locally compact groups |

22D25 | \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations |