New algebras of functions on topological groups arising from G-spaces

Abstract

Abstract. For a topological group G we introduce the algebra SU C(G) of strongly uniformly continuous functions. We show that SU C(G) contains the algebra W AP (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, we show that SU C(G) is trivial. We introduce the notion of fixed point on a class P of flows (P -f pp) and study in particular groups with the SU C-f pp. We study the Roelcke algebra (= U C(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 W AP (G) = SU C(G) = U C(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 SU C(G) is properly contained in U C(G). We then deduce that for this group U C(G) does not yield a right topological semigroup compactification.

Weakly almost periodic functions, model-theoretic stability, and minimality of topological groups

“…In [GM,Question 10.5], Glasner and Megrelishvili ask for the existence of a group which is WAP trivial but does not contain a copy of Homeo + ([0, 1]). In fact, Homeo(L) is such a group: by [BK], Homeo(L) is totally disconnected (and therefore does not contain a copy of Homeo + ([0, 1])) and by the above remark, it is WAP trivial.…”

“…That S ∞ is Roelcke precompact was first shown by Roelcke-Dierolf [RD]; the compactification was calculated by Uspenskij [U4] and Glasner-Megrelishvili [GM,Section 12].…”

Аменабельность Группы Подстановок Формальных Степенных Рядов

“…Если G -топологическая группа, то ЛИС на пространстве C * (G) соответствуют инвариантным вероятностным мерам на βG. Аналогично, для топологической группы G левоинвариантные средние на пространстве C * ru (G) соответствуют инвариантным вероятностным мерам на γ u G. Отметим, что Е. Гласнер и М. Г. Мегрелишвили изучали [21] и другие специальные эквивариантные компактные расширения группы, т. е. некоторые специальные эквивариантные подалгебры алгебры C * ru (G). Для основного рассматриваемого примера, группы J (k), где кольцо k бес-конечно, мы не можем утверждать, что инвариантные вероятностные меры на βJ (k) существуют.…”

Representations of dynamical systems on Banach spaces not containing $l_{1}$

“…The topological concept of fragmentability comes in fact from Banach space theory (Jayne-Rogers [23]). For dynamical applications of fragmentability, we refer to [28,29,30,13,14,15]. Fact 1.4 suggests the following general definition.…”

“…The material in this section is mostly well known. For more details and undefined concepts, see for example [13,12,14].…”