Recent work by Baum et al. [‘Expanders, exact crossed products, and the Baum–Connes conjecture’, Ann. K-Theory 1(2) (2016), 155–208], further developed by Buss et al. [‘Exotic crossed products and the Baum–Connes conjecture’, J. reine angew. Math. 740 (2018), 111–159], introduced a crossed-product functor that involves tensoring an action with a fixed action
$(C,\unicode[STIX]{x1D6FE})$
, 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
$(C,\unicode[STIX]{x1D6FE})$
is the action by translation on
$\ell ^{\infty }(G)$
, 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
$B(G)$
.
The $K$-theory of Gromov’s translation algebras and the amenability of discrete groups
