Hilbert C*-module independence

In this talk, we review some kinds of independence in setting of operator algebras. We introduce the notion of Hilbert $C^*$-module independence and show that it is a natural generalization of the notion of $C^*$-independence. Furthermore, we demonstrate that even in the case of $C^*$-algebras this concept of independence is new and has a nice characterization in terms of Hahn–Banach type extensions.
This talk is based on a joint work with R. Eskandari, J. Hamhalter, and V. M. Manuilov.
Keywords: Hilbert $C^*$-module; $C^*$-independence; state; determining element; module independence.