The slice approximating property and Figiel-type problem on unit spheres. (English) Zbl 1533.46014
The authors introduce the following notion: a real Banach space \(X\) has the slice approximating property (SAP) if, for every maximal convex subset \(C\) of the unit sphere \(S_X\) and every supporting functional \(f\) of \(C\), one has
\[
\lim_{\varepsilon\to 0}H(S(f,\varepsilon),C)=0.
\]
Here, \(H\) denotes the Hausdorff distance and
\[
S(f,\varepsilon)=\{x\in B_X:f(x)>1-\varepsilon\}
\]
is the slice of the unit ball \(B_X\) determined by \(f\) and \(\varepsilon>0\).
It is proved that all uniformly convex spaces, all almost CL-spaces and all spaces with the so-called Tingley property have the SAP.
The main result of the paper is the following. Let \(E\) be a generalized lush space with the SAP and \(F\) another Banach space. Let \(T:S_E \rightarrow S_F\) be an isometry such that \[ \mathrm{co}(-T(C)\cup T(-C))\subseteq S_F \] for every convex set \(C\subseteq S_E\), where \(\mathrm{co}\) denotes the convex hull.
Then \(T\) admits a Figiel operator, i.e., there is a linear operator \(Q:\overline{\mathrm{span}}(T(S_E)) \rightarrow E\) such that \(\|Q\|=1\) and \(Q\circ T=\mathrm{id}_{S_E}\).
It is proved that all uniformly convex spaces, all almost CL-spaces and all spaces with the so-called Tingley property have the SAP.
The main result of the paper is the following. Let \(E\) be a generalized lush space with the SAP and \(F\) another Banach space. Let \(T:S_E \rightarrow S_F\) be an isometry such that \[ \mathrm{co}(-T(C)\cup T(-C))\subseteq S_F \] for every convex set \(C\subseteq S_E\), where \(\mathrm{co}\) denotes the convex hull.
Then \(T\) admits a Figiel operator, i.e., there is a linear operator \(Q:\overline{\mathrm{span}}(T(S_E)) \rightarrow E\) such that \(\|Q\|=1\) and \(Q\circ T=\mathrm{id}_{S_E}\).
Reviewer: Jan-David Hardtke (Leipzig)
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