On amenable and coamenable coideals

We study relative amenability and amenability of a right coideal crete 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! ! c L degrees O. degrees O . G y / that generalizes coamenability of a quotient as defined by Kalantar, Kasprzak, Skalski, and Vergnioux (2022), where G y is the compact dual of G . In particular, we establish that the coamenable compact quasi-subgroups of G y are in one-to-one correspondence with the idempotent states on the reduced C *-algebra Cr r .Gy/. G y /. We use this work to obtain results for the duality between relative amenability and amenability of coideals in `degrees O.G/ degrees O . G / and coamenability of their codual coideals in L degrees O.Gy/, degrees O . G y /, making progress towards a question of Kalantar et al. N z P c `degrees O.G/ degrees O . G / of a dis