A reciprocity theorem for ergodic actions

Lange, Kenneth

doi:10.1090/S0002-9947-1972-0293004-0

A reciprocity theorem for ergodic actions
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Trans. Amer. Math. Soc. 167 (1972), 59-78 Request permission

Abstract:

An analogue of the Frobenius Reciprocity Theorem is proved for virtual groups over a locally compact separable group G. Specifically, an ergodic analytic Borel G-space $M(V\pi )$ is constructed from a virtual group V and a homomorphism $\pi :V \to G$ of V into G. This construction proves to be functorial for the category of virtual groups over G; in fact, it is a left adjoint of the functor which takes an ergodic analytic Borel G-space T into the virtual group $T \times G$ together with projection $\rho :T \times G \to G$ onto G. Examples such as Kakutani’s induced transformation and flows under functions show the scope of this construction. A method for constructing the product of two virtual groups is also presented. Some of the structural properties of the product virtual group are deduced from those of the components. Finally, for virtual groups ${\pi _1}:{V_1} \to {G_1}$ and ${\pi _2}:{V_2} \to {G_2}$ over groups ${G_1}$ and ${G_2}$ respectively, the adjoint functor construction applied to ${\pi _1} \times {\pi _2}:{V_1} \times {V_2} \to {G_1} \times {G_2}$ is shown to give the product of the ${G_1}$-space derived from ${\pi _1}:{V_1} \to {G_1}$ and the ${G_2}$-space derived from ${\pi _2}:{V_2} \to {G_2}$, up to suitably defined isomorphism.

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