Lipschitz and uniform embeddings into $\ell _{\infty} $
Volume 212 / 2011
Fundamenta Mathematicae 212 (2011), 53-69
MSC: 46B20, 46B25, 46T99.
DOI: 10.4064/fm212-1-4
Abstract
We show that there is no uniformly continuous selection of the quotient map $Q:\ell _\infty \to \ell _\infty //c_0$ relative to the unit ball. We use this to construct an answer to a problem of Benyamini and Lindenstrauss; there is a Banach space $X$ such that there is a no Lipschitz retraction of $X^{**}$ onto $X$; in fact there is no uniformly continuous retraction from $B_{X^{**}}$ onto $B_X$.
