Uniform ergodic theorems for semigroup representations

We consider a bounded representation T of a commutative semigroup S on a Banach space and analyse the relation between three concepts: (i) properties of the unitary spectrum of T, which is defined in terms of semigroup characters on S; (ii) uniform mean ergodic properties of T; and (iii) quasi-compactness of T. We use our results to generalize the celebrated Niiro-Sawashima theorem to semigroup representations and, as a consequence, obtain the following: if a positive and bounded semigroup representation on a Banach lattice is uniformly mean ergodic and has finite-dimensional fixed space, then it is quasi-compact.