MR2543732 (2010g:46038) Colao, Vittorio; Trombetta, Alessandro; Trombetta, Giulio Hausdorff norms of retractions in Banach spaces of continuous functions. Taiwanese J. Math. 13 (2009), no. 4, 1139–1158. (Reviewer: Diana Caponetti)

Abstract

A retraction $R$ from the closed unit ball of a Banach space $X$ onto its boundary is called $k$-ball contractive if there is $k \ge 0$ such that $ \gamma_X(RA) \le k \gamma_X(A) $ for each subset $ A$ of the closed unit ball, where $\gamma_X$ denote the Hausdorff (ball) measure of noncompactness. In the paper under review the authors consider the problem of evaluating the Wo\'{s}ko constant, which is the infimum of all numbers $k$'s for which there is a $k$-ball contractive retraction from the closed unit ball onto the sphere, in Banach spaces of real continuous functions defined on domains which are not necessarily bounded or finite dimensional. The paper extends some previous results valid in spaces of continuous functions to a more general setting. The authors consider the space $\mathcal{B}\mathcal{C}_{B(E)}(K)$ of all real bounded functions which are continuous on $K$ and uniformly continuous on the closed unit ball $B(E)$, being $E$ a normed space and $K$ a set in $E$ containing $B(E)$. They also consider the space ${\mathcal C}(P)$ of all real continuous functions defined on the Hilbert cube $P= \{ x=(x_n) \in l_2 : |x_n| \le \frac1{n} \ \ (n=1,2, ...) \}$. They prove that in both the spaces $\mathcal{B}\mathcal{C}_{B(E)}(K)$ and ${\mathcal C}(P)$ the Wo\'{s}ko constant assumes the smallest possible value $1$, they also give precise estimates of the lower Hausdorff norms and the Hausdorff norms of the retractions they construct.