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A basic result in equivariant topology, due to Greenlees and May, completely determines the structure of rational genuine G-spectra for a finite group G: they form a semisimple abelian tensor category, equivalent to that of rational G-Mackey functors, which itself splits into a product of categories of rational representations of the Weil groups of subgroups of G. In this talk, I will explain how to generalize the algebraic splitting from the Burnside ring to any Green functor, and how this can be applied to equivariant KK-theory and to the homotopy category of suitable equivariant S-algebras.