  We prove the following two results. First, the isometry semigroup of a unital
properly infinite nuclear C*-algebra is right amenable. Second, the unitary
group of a unital simple monotracial C*-algebra whose tracial GNS
representation is hyperfinite is skew-amenable in the weak topology. This
answers in part a conjecture of Alekseev, Schmidt, and Thom and a question of
Pestov.
