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

E. Glasner, and M. Megrelishvili. "New algebras of functions on topological groups arising from G-spaces." Fundamenta Mathematicae 201.1 (2008): 1-51. <http://eudml.org/doc/282828>.

@article{E2008,
abstract = {For a topological group G we introduce the algebra SUC(G) of strongly uniformly continuous functions. We show that SUC(G) 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, we show that SUC(G) is trivial. We introduce the notion of fixed point on a class P of flows (P - fpp) and study in particular groups with the SUC-fpp. 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), 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). We then deduce that for this group UC(G) does not yield a right topological semigroup compactification.},
author = {E. Glasner, M. Megrelishvili},
journal = {Fundamenta Mathematicae},
keywords = {Asplund function; fixed point property; -compactification; -space; locally equicontinuous; matrix coefficient; right topological semigroup compactification; strongly uniformly continuous},
language = {eng},
number = {1},
pages = {1-51},
title = {New algebras of functions on topological groups arising from G-spaces},
url = {http://eudml.org/doc/282828},
volume = {201},
year = {2008},
}

TY - JOUR
AU - E. Glasner
AU - M. Megrelishvili
TI - New algebras of functions on topological groups arising from G-spaces
JO - Fundamenta Mathematicae
PY - 2008
VL - 201
IS - 1
SP - 1
EP - 51
AB - For a topological group G we introduce the algebra SUC(G) of strongly uniformly continuous functions. We show that SUC(G) 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, we show that SUC(G) is trivial. We introduce the notion of fixed point on a class P of flows (P - fpp) and study in particular groups with the SUC-fpp. 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), 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). We then deduce that for this group UC(G) does not yield a right topological semigroup compactification.
LA - eng
KW - Asplund function; fixed point property; -compactification; -space; locally equicontinuous; matrix coefficient; right topological semigroup compactification; strongly uniformly continuous
UR - http://eudml.org/doc/282828
ER -