Tensor-Product Coaction Functors

Abstract

Recent work by Baum, Guentner, and Willett, and further developed by Buss, Echterhoff, and Willett introduced a crossed-product functor that involves tensoring an action with a fixed action (C, γ), then forming the image inside the crossed product of the maximal-tensor-product action. For discrete groups, we give an analogue, for coaction functors. We prove that composing our tensor-product coaction functor with the full crossed product of an action reproduces their tensor-crossed-product functor. We prove that every such tensor-product coaction functor is exact; and if (C, γ) is the action by translation on ℓ ∞ (G), we prove that the associated tensor-product coaction functor is minimal, thereby recovering the analogous result by the above authors. Finally, we discuss the connection with the E-ization functor we defined earlier, where E is a large ideal of B(G).