On Amenable and Coamenable Coideals

We study relative amenability and amenability of a right coideal ${\tilde{N}}_P\subseteq {\mathrm{\ell }}^{\mathrm{\infty }}(\mathbb{G})$ of a discrete quantum group in terms of its group-like projection $P$. We establish a notion of a $P$-left invariant state and use it to characterize relative amenability. We also develop a notion of coamenability of a compact quasi-subgroup $N_{\omega }\subseteq L^{\mathrm{\infty }}(\widehat{\mathbb{G}})$ that generalizes coamenability of a quotient as defined by Kalantar, Kasprzak, Skalski, and Vergnioux, where $\widehat{\mathbb{G}}$ is the compact dual of $\mathbb{G}$. In particular, we establish that the coamenable compact quasi-subgroups of $\widehat{\mathbb{G}}$ are in one-to-one correspondence with the idempotent states on the reduced $C^{\ast }$-algebra $C_r(\widehat{\mathbb{G}})$. We use this work to obtain results for the duality between relative amenability and amenability of coideals in ${\mathrm{\ell }}^{\mathrm{\infty }}(\mathbb{G})$ and coamenability of their codual coideals in $L^{\mathrm{\infty }}(\widehat{\mathbb{G}})$, making progress towards a question of Kalantar et al{.}.