Itzkowitz's problem for groups of finite exponent

Ahmed Bouziad (1) , Aicha Bareche (1)

1 LMRS - Laboratoire de Mathématiques Raphaël Salem

Itzkowitz's problem asks whether every topological group $G$ has equal left and right uniform structures provided that bounded left uniformly continuous real-valued function on $G$ are right uniformly continuous. This paper provides a positive answer to this problem if $G$ is of bounded exponent or, more generally, if there exist an integer $p\geq 2$ and a nonempty open set $U\subset G$ such that the power map $U\ni g\to g^p\in G$ is left (or right) uniformly continuous. This also resolves the problem for periodic groups which are Baire spaces.

Mots clés: Topological group; Left (right) uniform structure; Left (right) uniformly continuous bounded real-valued function; SIN-group; FSIN-group; Left (right) thin subset; Left (right) neutral subset; Left (right) uniformly discrete subset; Group of finite exponent

Domaines: Topologie générale [math.GN]; Théorie des groupes [math.GR]

Ahmed Bouziad, Aicha Bareche. Itzkowitz's problem for groups of finite exponent. 2009. ⟨hal-00464999⟩