Existence of ergodic retractions for semigroups in Banach spaces

In this work, we prove among other results that if S is a right amenable semigroup and φ={Ts:s∈S} is a (quasi-)nonexpansive semigroup on a closed, convex subset C in a strictly convex reflexive Banach space E such that the set F(φ) of common fixed points of φ is nonempty, then there exists a (quasi-)nonexpansive retraction P from C onto F(φ) such that PTt=TtP=P for each t∈S and every closed convex φ-invariant subset of C is also P-invariant. Moreover, if the mappings are also affine then Tμ [G. Rode, An ergodic theorem for semigroups of nonexpansive mappings in a Hilbert space, J. Math. Anal. Appl. 85 (1982) 172-178. [12]] is a quasi-contractive affine retraction from C onto F(φ), such that TμTt=TtTμ=Tμ for each t∈S, and Tμx∈co¯{Ttx:t∈S} for each x∈C; and if R is an arbitrary retraction from C onto F(φ) such that Rx∈co¯{Ttx:t∈S} for each x∈C, then R=Tμ. It is shown that if the Tt's are F(φ)-quasi-contractive then the results hold without the strict convexity condition on E.