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Nonexpansive retracts in Banach spaces. (English) Zbl 1125.46019

Jachymski, Jacek (ed.) et al., Fixed point theory and its applications. Proceedings of the international conference, Bȩdlewo, Poland, August 1–5, 2005. Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Center Publications 77, 161-174 (2007).
A nonexpansive retract of a Banach space or a metric space is the range of a nonexpansive retraction. A space has the compact nonexpansive envelope property if each compact subset is contained in a compact nonexpansive retract. It is shown that \(c_0\) has this property while \(\ell_p\) lacks it when \(1\leq p<\infty\) and \(p\neq2\); other related examples and counterexamples are given. It is also shown that the unit ball of \(\ell_1\) is not a nonexpansive retract (for \(\ell_p\), \(1<p<\infty\), \(p\neq2\), this follows from a result of R. E. Bruck [Proc. Am. Math. Soc. 43, 173–175 (1974; Zbl 0306.46027)]). Finally, it is shown that in many Banach spaces the nonexpansive retracts are even the range of a “sunny” nonexpansive retraction \(R\), i.e.,\(R(v+r(x-v))=v\) whenever \(Rx=v\) and \(r\geq0\).
For the entire collection see [Zbl 1112.47302].

MSC:

46B99 Normed linear spaces and Banach spaces; Banach lattices
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
52A05 Convex sets without dimension restrictions (aspects of convex geometry)
52A55 Spherical and hyperbolic convexity

Citations:

Zbl 0306.46027
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