The transfer of property \((\beta )\) of Rolewicz by a uniform quotient map (Q2790729)

It is proved that the property of having an equivalent norm with the property \((\beta)\) of Rolewicz is qualitatively preserved via surjective uniform quotient mappings between separable Banach spaces.NEWLINENEWLINEAn important part of the proof is the following metric characterization: A separable reflexive Banach space \(Y\) has an equivalent norm with property \((\beta)\) if and only if it does not contain any bi-Lipschitz copy of the metric tree \(\mathbb{T}_\infty\) if all finite subsets of \(\mathbb{N}\) with the shortest path metric. A graph \(\mathbb{M}_\infty\), defined after a variant of the Laakso construction and a map from \(\mathbb{T}_\infty\) to \(\mathbb{M}_\infty\) and a fork argument is then used to show the reflexive case. The general case uses a characterization of non-reflexivity of James.NEWLINENEWLINEIt is also shown that the \((\beta)\)-modulus is not quantitatively preserved via such surjective quotient maps by exhibiting two uniformly homeomorphic Banach spaces that do not have \((\beta)\)-moduli of the same power type under renorming.