On the semigroup structure of state spaces of non-commutative unitary C*-algebras

We study Green's relations on the state space of a certain class of compact quantum semigroups, namely non-commutative unitary C*-algebras. Furthermore, we generalize Friedberg's results on finite-dimensional representations of group-extremal affine semigroups to the state space of noncommutative unitary C*-algebras. This yields new examples of compact semigroups that admit a well-behaved finite-dimensional representation theory beyond the group-extremal setting.