  A notion of unique ergodicity relative to the fixed-point subalgebra is
defined for automorphisms of unital C*-algebras. It is proved that the free
shift on any reduced amalgamated free product C*-algebra is uniquely ergodic
relative to its fixed-point subalgebra, as are autormorphisms of reduced group
C*-algebras arising from certain automorphisms of groups. A generalization of
Haagerup's inequality, yielding bounds on the norms of certain elements in
reduced amalgamated free product C*-algebras, is proved.
