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I am interested in Pontryagin duality-like theories for discrete groups, more particularly, whether an analogue to Bochner's theorem for abelian groups exists in the discrete non-finite and non-abelian setting.

I have seen the paper "Lubotzky - Tannaka duality for discrete groups" and know about the Hopf-algebra generalizations of the Pontryagin dual but neither reference a Bochner's theorem, so my question is:

Is there a duality theory for a discrete (amenable) group and an analogue of Bochner's theorem in such a theory?

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