Itzkowitz's problem for group of finite exponent

TL;DR

This paper addresses Itzkowitz's problem in topological groups, providing positive results for groups of bounded exponent or with specific uniform continuity properties, and resolves the problem for periodic Baire groups.

Contribution

It offers a positive solution to Itzkowitz's problem for groups of bounded exponent and certain periodic groups with specific uniform continuity conditions.

Findings

Positive answer for groups of bounded exponent

Resolution for periodic Baire groups

Conditions involving power maps and uniform continuity

Abstract

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.