A Characterization of Amenable Locally Compact Quantum Groups

In this paper, we recall the notion of locally compact quantum groups and character injective Banach modules from the current literature. Then, we show that a locally compact quantum group $\mathbb{G}$ is amenable if and only if each $J\in \mathbf{mod-L^1(\mathbb{G})}$ is $y$-flat for $y\in \text{sp} (L^1(\mathbb{G}))$. $\sec $