As is well known, the equivalence between amenability of a locally compact group G and injectivity of its von Neumann algebra L(G) does not hold beyond inner-amenable groups. We show that the equivalence persists for all locally compact groups if L(G) is considered as a T(L_2(G))-module with respect to a natural action, which leads us to the notion of covariant injectivity. In fact, we present an appropriate version of this result for every locally compact quantum group H. This gives rise to several characterizations of quantum group amenability in terms of injectivity in the category of T(L_2(H))-modules.

This is joint work with my former PhD student Jason Crann (TAMS 2016).