源于G-空间的拓扑群上的新函数代数
New algebras of functions on topological groups arising from G-spaces
AI总结：
引入拓扑群G上的强一致连续函数代数SUC(G)，研究其包含关系与紧致化，并证明具有可度量化M(G)的Polish群的SUC-顺从性与极端顺从性。
摘要：
对于拓扑群G，我们引入了强一致连续函数的代数SUC(G)。它包含弱概周期函数的代数WAP(G)，以及局部等度连续函数和Asplund函数的代数LE(G)与Asp(G)。对于单位区间保序同胚群和直径为1的Urysohn空间等距群的Polish群，SUC(G)是平凡的。我们研究了可数集置换群S(N)和Cantor集同胚群H(C)的Roelcke代数（=UC(G)=左右一致连续函数）及SUC紧致化。对于第一个群，我们证明WAP(G)=SUC(G)=UC(G)，并给出了相应的可度量化（实际上是Cantor）半拓扑半群紧致化的具体描述。对于第二个群，我们证明SUC(G)真包含于UC(G)，且UC(G)不产生右拓扑半群紧致化。我们引入了在流类P上的不动点概念（P-fpp），特别研究了SUC-顺从群和具有SUC-fpp的群（SUC-极端顺从性）。我们证明每个具有可度量化M(G)的Polish群G都是SUC-顺从的，且若M(G)是近距的，则G是SUC-极端顺从的。
英文摘要：
For a topological group G we introduce the algebra SUC(G) of strongly uniformly continuous functions. It contains the algebra WAP(G) of weakly almost periodic functions as well as the algebras LE(G) and Asp(G) of locally equicontinuous and Asplund functions respectively. For the Polish groups of order preserving homeomorphisms of the unit interval and of isometries of the Urysohn space of diameter 1, SUC(G) is trivial. We study the Roelcke algebra (= UC(G) = right and left uniformly continuous functions) and SUC compactifications of the groups S(N), of permutations of a countable set, and H(C), the group of homeomorphisms of the Cantor set. For the first group we show that WAP(G)=SUC(G)=UC(G) and also provide a concrete description of the corresponding metrizable (in fact Cantor) semitopological semigroup compactification. For the second group, in contrast, we show that SUC(G) is properly contained in UC(G) and for this group UC(G) does not yield a right topological semigroup compactification. We introduce the notion of fixed point on a class P of flows (P-fpp) and study in particular groups which are SUC-amenable and groups with the SUC-fpp (SUC-extreme amenability). We show that every Polish group G with metrizable M(G) is SUC-amenable and if, in addition, M(G) is proximal, then G is SUC-extremely amenable.
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