In view of Takesaki-Takai duality, we can go back and forth between C*-dynamical systems of an abelian group and ones of its Pontryagin dual by taking crossed products. In this talk, I present a similar duality between actions on C*-algebras of two constructions of locally compact quantum groups: one is the bicrossed product due to Vaes-Vainerman, and the other is the double crossed product due to Baaj-Vaes. I will explain the situation by illustrating the example coming from groups. If time permits, I will also discuss its consequences in the case of quantum doubles.

K-homology is the dual theory to K-theory for C*-algebras.  I will show how under appropriate quasi-diagonality and countability assumptions K-homology (more generally, KK-theory) can be realized by completely positive and contractive, and approximately multiplicative, maps to matrix algebras modulo an appropriate equivalence relation.  I’ll briefly explain some connections to manifold topology and existence / uniqueness theorems in C*-algebra classification theory (due to Dadarlat and Eilers).

Some of this is based on joint work with Guoliang Yu, and some is work in progress