Subproduct systems, Gysin sequences and SU(2)-symmetries

For a $C^*$-correspondence from a $C^*$-algebra to itself one may associate a $C^*$-algebra referred to as the Cuntz–Pimsner algebra of the $C^*$-correspondence. Special cases are the Cuntz–Krieger algebras and crossed products by the integers. Furthermore, the $K$-theory of Cuntz–Pimsner algebras can often be computed by means of a six term exact sequence which generalizes the $K$-theoretic Gysin sequence of a complex hermitian line bundle.
A more general construction of $C^*$-algebras associated to module theoretic data comes from subproduct systems over the monoid of non-negative integers. But so far in this context there are no general tools available for computing the $K$-groups of such a Cuntz–Pimsner algebra.
In this talk we investigate a class of $C^*$-algebras constructed from a finite dimensional representation of SU(2) via an associated subproduct system. We compute the $K$-theory of this kind of Cuntz–Pimsner algebra by means of a six term exact sequence sharing the characteristic properties of the $K$-theoretic Gysin sequence of a complex hermitian vector bundle of rank 2.
The talk is based on joint work with Francesca Arici.