Sequential criteria for equicontinuity and uniformities on topological groups

Abstract

It was asked by Itzkowitz in 1976 whether or not the equality of left and right uniformities on a locally compact group G could be decided by sequences. Twelve years later, the question was answered in the affirmative: For the left and right uniformities on G to be equal, it is necessary and sufficient that for each sequence ( h n ) in G and each neighborhood V of the identity e in G, the set ∩ nϵN h n −1 Vh n should be a neighborhood of e. Two independent proofs of this unexpected result were proposed, one in the spirit of set-theoretic topology (Pestov), and the other based on finite-dimensional Lie theory (Itzkowitz, Rothman, Strassberg and Wu). Shortly after, Hansel and Troallic referred to techniques of harmonic analysis to prove that a symmetric set B of automorphisms of G is equicontinuous if and only if the countable subsets of B are equicontinuous ( G being endowed with its left uniformity). This substantially betters the aforementioned property, which can be obtained if B is chosen as the set of all inner automorphisms of G. In this paper, we re-examine the result and consider it in a maximal generality. We obtain the following basic property: Let X be a quasi-k-space (for example a locally compact space), Y a uniform space and H a subset of C (X,Y); then H is equicontinuous if and only if for each countable subset A of X and each countable subset D of H , the set D | A of restrictions to A of mappings of D is equicontinuous. Several applications of this criterion are given, among which a strengthened version of a well-known theorem of Corson and Glicksberg. In the last section, the technique is applied to the sets of inner automorphisms of topological groups, and a refinement of the result by Itzkowitz, Pestov, Rothman, Strassberg and Wu is obtained.