Multi-norms and the injectivity of $L^p(G)$
  Let $G$ be a locally compact group, and take $p\in(1,\infty)$. We prove that the Banach left $L^1(G)$-module $L^p(G)$ is injective (if and) only if the group $G$ is amenable. Our proof uses the notion of multi-norms. We also develop the theory of multi-normed spaces. 
H. Garth Dales, Matthew Daws, Hung Le Pham, Paul Ramsden
math.FA