Constant functions and left invariant means on semigroups

Abstract

semigroup, is left amenable, then K contains a common fixed point of the fanmily S. Proof. Since S is left amenable, then by Theorem 3, there exists a net, {T1} where T. E P, such that for any fe En(S), {T,,f} converges pointwise to a constant function. For any T,, there exists 0, E 'D such that T, = E, s4,(s)r.. Let the map J,: KK be given by J,,k = Y, E s4,(s)s(k), for k E K. For the remainder of the proof, let y be a specific point in K. By compactness of K, there exists a subnet, {J,} of {J,}, such that {J,6y} converges to some yo E K. And the associated subnet, {T,} of {T}, satisfies that for anyfe rm(S), {Tjf} converges pointwise to a constant function, since {TJf} is a subnet of {Tj}. We will show that for any so E S, that soyo is the required common fixed point. (It can be shown by counterexample that yo itself need not be one.) For each tE X*, define as in [2, p. 587], a real valued function, f,E on S byf,E(s) = ,(sy), for s E S. Since ,u is continuous over the compact set K, then f,E e m(S). So for s' eS, (TJ ,1)s' = OAS) ((rsf) ) = O ?As)f(s 's) = E A(s)p(s 'sy) s e S s e S (s ( S SY) =l(S (sS(S)s)) = 1(s'J6y), by linearity of ,u and affineness of s', in steps 4 and 5, respectively. Then lim (T,6f,)s' = lim (r(s'(J6y))) = ,t(s' ( lim (J,6Y)))