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On topological semigroups with invariant means in the convex hull of multiplicative means Lau, Anthony To-Ming

Abstract

Let S be a topological semigroup and C(S) the space of bounded real continuous function on S with sup norm. For f є C(S), s, t є S, [subscript s]f(t) = f(st), let LUC(S) be the space of all f є C(S) for which the map s ↦ [subscript s] f from S to C(S) is continuous, Δ(S) the set of multiplicative means on LUC(S), and CoΔ(S) the convex hull of Δ(S). In this thesis we study and characterize topological semigroup S for which (*) LUC(S) has a LIM (left invariant mean) in CoΔ(S). A decomposition theorem for such semigroups has been obtained. We also consider properties that arise from the action of semigroups satisfying (*) on certain topological spaces. In particular, we generalise Mitchell's fixed point theorem (theorem 1 [28]). Other characterization theorems and combinatorial properties for such semigroups have also been obtained. Continuing the work of J. Sorenson [32], we obtain characterizations and functional analytic properties for discrete semigroups satisfying (*), generalising some of the results of Granirer [12], [13], [l14] and Mitchell [26] for semigroups admitting a multiplicative LIM. Finally we characterize all semigroups S for which m(S) (the space of hounded real functions) has a non-trivial translation invariant subalgebra, containing constants and admits a multiplicative LIM. We also give a method, utilizing the class of left thick subsets of Mitchell [25], in constructing a huge class of such subalgebras. Furthermore, we show that the above method and characterization is valid even for semigroup of transformations. Other diverse results in this direction are also obtained.

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