Duality of restriction and induction for 𝐶*-coactions

Kaliszewski, S.; Quigg, John; Raeburn, Iain

doi:10.1090/S0002-9947-97-01905-3

Duality of restriction and induction for $C^*$-coactions
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Trans. Amer. Math. Soc. 349 (1997), 2085-2113 Request permission

Abstract:

Consider a coaction $\delta$ of a locally compact group $G$ on a $C^*$ algebra $A$, and a closed normal subgroup $N$ of $G$. We prove, following results of Echterhoff for abelian $G$, that Mansfield’s imprimitivity between $A\times _{\delta |}G/N$ and $A\times _\delta G\times _{\hat \delta ,r}N$ implements equivalences between Mansfield induction of representations from $A\times _{\delta |}G/N$ to $A\times _\delta G$ and restriction of representations from $A\times _\delta G\times _{\hat \delta ,r}N$ to $A\times _\delta G$, and between restriction of representations from $A\times _\delta G$ to $A\times _{\delta |}G/N$ and Green induction of representations from $A\times _\delta G$ to $A\times _\delta G\times _{\hat \delta ,r}N$. This allows us to deduce properties of Mansfield induction from the known theory of ordinary crossed products.

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