Anthony R. Weston Canisius College (USA)
Analysis of group structures induced by uniform homeomorphisms
Uniform homeomorphisms—that is to say, bijections which are uniformly continuous in both directions—induce
group structures on complete normed vector spaces (Banach spaces) that “resemble” the additive structures of
the underlying spaces.
Analysis of uniform Banach groups sheds new light on open questions about the classification of Banach spaces
up to uniform homeomorphisms. For example, in this talk, we will detail a very general answer to a question
recently highlighted by Yoav Benjamini and Joram Lindenstrauss in their monograph Geometric Nonlinear
Functional Analysis (Volume 1): Can a nonnormable topological vector space be uniformly homeomorphic to a
Banach space?
Our approach to this question, via uniform Banach groups, determines a new linearization procedure for a wide
class of uniform homeomorphisms. Such techniques are rare because uniformly continuous maps do not have
derivatives in general, rendering affine approximation difficult.
