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On topological semigroups with invariant means in the convex hull of multiplicative means

University of British Columbia

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.

Text

2011-06-21

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http://hdl.handle.net/2429/35609

Mathematics

University of British Columbia

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