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?
Best Answer
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.