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 in general beyond inner amenable groups. In this paper, we show that the equivalence persists for all locally compact groups if L(G) is considered as a T (L2(G))-module with respect to a natural action. In fact, we prove an appropriate version of this result for every locally compact quantum group.

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Journal Transactions of the American Mathematical Society
Crann, J, & Neufang, M. (2016). Amenability and covariant injectivity of locally compact quantum groups. Transactions of the American Mathematical Society, 368(1), 495–513. doi:10.1090/tran/6374