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 phi = {T-s : s E 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(phi) of common fixed points of phi is nonempty, then there exists a (quasi-)nonexpansive retraction P from C onto F(phi) such that PTt = T-t P = P for each t is an element of S and every closed convex phi-invariant subset of C is also P-invariant. Moreover, if the mappings are also affine then T-mu [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(phi), such that T mu Tt = T-t T-mu = T-mu for each t is an element of S, and T(mu)x is an element of (co) over bar {T(t)x : t is an element of S} for each x is an element of C; and if R is an arbitrary retraction from C onto F(phi) such that Rx is an element of (co) over bar {T(t)x: t is an element of S} for each x is an element of C, then R = T-mu. It is shown that if the T-t's are F(phi)-quasi-contractive then the results hold without the strict convexity condition on E. (c) 2007 Elsevier Ltd. All rights reserved.