Noncommutative join construction

The aim of this talk is to show how to carry out the join construction of compact quantum groups avoiding braiding and replacing the unit interval by an arbitrary unital C*-algebra (noncommutative compact Hausdorff space). This is done in terms of equivariantly projective Hopf-Galois extensions of C*-algebras. The completion of the extended algebra is a natural candidate for a non-crossed product example of a principal extension of C*-algebras in the sense of Ellwood (non-trivial noncommutative principal bundle). The main point is a general and explicit formula for a strong connection, which puts us directly into the framework of the index pairing between K-theory and K-homology. (Based on a joint work with L. Dąbrowski and T. Hadfield.) Piotr M. Hajac (Instytut Matematyczny, Polska Akademia Nauk / Katedra Metod Matematycznych Fizyki, Uniwersytet Warszawski). Joint Noncommutative Geometry and Algebraic Topology Seminar.