C*-algebras with quantum group symmetry: Noncommutative boundaries and equivariant subproduct systems

Abstract

Mathematical objects often possess some sort of 'symmetry'. For instance, the circle has rotational symmetry; It 'looks the same' even if we rotate it around its center by some angle. Formally this type of symmetry is an example of a group action on a topological space. Groups can also act on C*-algebras (sometimes called 'quantum spaces'), which are objects that generalize topological spaces. However, in this setting it is interesting to in addition look at 'quantum symmetries'. These are encoded by quantum group actions, which is the overarching topic of the thesis. On the one hand we consider C*-algebras constructed from certain polynomials, via so-called subproduct systems. These turn out to posess quantum symmetry, something we use both to describe the C*-algebras and to study equivariant KK-theory. The descriptions of the C*-algebras also shed light on representation theory and connections to braided quantum groups. On the other hand we can start with a quantum group, and consider so-called 'noncommutative boundaries'. Given a compact quantum group we show that its Drinfeld double always has a Furstenberg–Hamana boundary. This is a universal object which is often closely related to the Poisson boundaries.