Fixed Point and Finite Dimensional Invariant Subspace Properties for Semigroups and Amenability

A well-known theorem of Markov-Kakutani [5, p. 456] asserts that if S is a commutative semigroup, then S has the following fixed point property: (1) whenever S = {T _S ; s ∈ S} is a representation of S as affine continuous mappings from a non-empty compact convex subset K of a separated locally covex space (i.e. T _s (λx + (1 − λ)y) = λT _s (x) + (1 − λ)T _s (y), 0 ≤ λ ≤ 1, x, y ∈ K), then K contains a common fixed point for S.