In the present paper, the author continues some of his recent joint works with W. Takahashi on fixed point properties or ergodic properties for semigroups of nonexpansive mappings on a nonempty closed convex subset of a Banach space. Especially, the author studies the algebra of (nonlinear) submeans on subspaces of \(l^\infty(S)\), where \(S\) is a semigroup and \(l^\infty(S)\) is the Banach space of bounded and real valued function on \(S\) with the supremum norm, the relationship between invariant means (or submeans) and fixed point properties of semigroups of nonexpansive mappings, the relation between amenability and ergodic type theorems, and the approximation of fixed points.