We describe how the constructions of quantum homogeneous spaces using infinitesimal invariance and quantum coisotropic subgroups are related. As an example we recover the quantum 4-sphere of [2] through infinitesimal invariance with respect to ${}_q\left(SU\left(2\right)\right)$.

We review the formulation and proof of the Baum-Connes conjecture for the dual of the quantum group $S{U}_q\left(2\right)$ of Woronowicz. As an illustration of this result we determine the K-groups of quantum automorphism groups of simple matrix algebras.