Some New Algebras of Functions on Topological Groups Arising from G-spaces

We investigate new and old algebras of continuous functions on topological groups G arising from G-spaces and some associated linear representations G → Iso (V ) on Banach spaces V . The topological group G = H+[0, 1] of orientation preserving homeomorphisms on the closed interval does not admit non-trivial continuous representations G → Iso (V ) when V is a reflexive Banach space (an equivalent statement is that every weakly almost periodic function on G is constant), [27]. We strengthen this result by showing that the same is true when V is an Asplund Banach space. In turn, this statement is equivalent to the triviality of the algebra Asp(G) of Asplund functions on G. For an arbitrary topological group G we introduce a new algebra of functions, the algebra of strongly uniformly continuous functions which appears to be a useful tool in the study of non-abelian groups. We show that for any G the inclusions RUC(G) ∩ LUC(G) ⊃ SUC(G) ⊃ LE(G) ⊃ Asp(G) ⊃ WAP (G) hold (where LE(G) is the algebra of locally equicontinuous functions in the sense of Glasner and Weiss [15]) and that for G = H+[0, 1], and also for G = Iso (U1) the isometry group of U1 (the Urysohn space of diameter 1) we have SUC(G) = {constants}. We introduce the notion of fixed point on a class P of flows (P-fpp or extreme P-amenability) and study in particular groups with the SUC-fpp, namely those groups which have a fixed point on compact SUC-flows. Several of the well known large Polish groups which fail to be extremely amenable are extremely SUC-amenable. In the final sections we study the Roelcke and SUC compactifications of the group S∞ of permutations of N, and H(C), the group of homeomorphisms of the Cantor set. For the first group we show that WAP (S∞) = SUC(S∞) = UC(S∞) and also provide a concrete description of the corresponding metrizable (in fact Cantor) semitopological semigroup compactification. For the latter group, in contrast, we have SUC(H(C)) $ UC(H(C)) from which fact we deduce that UC(H(C)) is not a right topological semigroup compactification of H(C).