Let $\mathbb{G}$ be a discrete quantum group (in the sense of Vaes-Kustermans) with function algebra $(\ell^\infty(\mathbb{G}), \Delta)$. A (right) action of $\mathbb{G}$ on a von Neumann algebra $M$ is an injective, unital, normal $\*$-homomorphism $\alpha: M \to M \otimes \ell^\infty(\mathbb{G})$ such that
$$(\iota \otimes \Delta)\alpha = (\alpha \otimes \iota)\alpha.$$


Are there any interesting examples of (genuine) discrete quantum groups (i.e. not classical discrete groups) acting on **commutative** von Neumann algebras?



Examples of discrete quantum group actions on commutative von Neumann algebras
The dual coideals for Podleś sphere algebras are commutative coideal subalgebras of $\ell^\infty \widehat{\mathit{SU}}\_q(2)$. They can be thought as ‘quantized’ $L(T)$ for the 1-dimensional toruses $T < \mathit{SU}(2)$ sitting as coisotropic subgroups.


