Generalized amenability in quantum groups

We study generalized amenability of locally compact quantum groups $\mathbb{G}=(\text{scr}M,\mathrm{\Delta },\phi ,\psi )$. Let $C_r^{\ast }(\mathbb{G})$ and $VN(\mathbb{G})$ denote, respectively, the C*-algebra and von Neumann algebra generated by the quantum left regular representation of $\mathbb{G}$. With natural covariance assumptions and a traceability condition on $VN(\mathbb{G})$, we show that if $C_r^{\ast }(\mathbb{G})$ is approximately amenable, then $\mathbb{G}$ admits an invariant quantum mean. We show further that the same conclusion holds if the predual algebra $\text{scr}{M}_{\ast }$ has a bounded approximate identity and is approximately amenable.