Uniform structures and uniformly continuous functions on topological groups and their factor spaces

Abstract

A topological group is called a SIN group if the left and right uniform structures on G coincide, and a FSIN group (functionally SIN) if every bounded left uniformly continuous real valued functions on G is right uniformly continuous. Do the classes of SIN groups and of functionally SIN (FSIN) groups coincide? This question, belonging to G.Itzkowitz, is still open. It was answered in the affirmative for some particular classes of topological groups, including locally compact groups, metrisable groups, locally connected groups, and some other. We will show that the answer is positive for locally chain connected groups. More generally, Itzkowitz's question can be asked not only for topological groups but for their homogeneous factor spaces, because they are also known to support two natural uniform structures (although not necessarily compatible with the factor topology), the left and the right ones. Do these two uniformities in G/H coincide as soon as every left uniformly continuous real valued function on G/H is right uniformly continuous? We show that the answer to this question is in the negative.