Invariant means on locally compact semigroups

Abstract

Let G be a locally compact semigroup (jointly continuous semigroup operation), M(G) the algebra of all bounded regular Borel measures on G (with convolution as multiplication), E a separated locally convex space and S a compact convex subset of E. We show that there is a left invariant mean on the space LUC(G) of all bounded left uniformly continuous functions on G iff G has the following fixed point property: For any bilinear mapping T: M(G) x E E (denoted by (,a, s) -* T,,(s)) such that (a) T,,(S) C S for any It > O, Il,ull = 1, (b) TMu*v = Tu ? Tv for any It, vCM(G), (c) T,u:S-S is continuous for any ,u _ O, I,ul = 1, and (d) It TMu(s) is continuous for each s C S when M(G) has the topology induced by the seminorms pf(,U) = Sf f dyl, f C LUC(G), there is some so C S such that TM(so) = so for any I > 0, Ilufl = 1.