We provide a novel construction of quantized universal enveloping $*$-algebras of real semisimple Lie algebras, based on Letzter’s theory of quantum symmetric pairs. We show that these structures can be ‘integrated’, leading to a quantization of the group C${}^{*}$-algebra of an arbitrary semisimple algebraic real Lie group.

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.