Generalized-lush spaces and the Mazur-Ulam property (Q2866762)

The well-known Mazur-Ulam theorem states that every surjective isometry \(T:X \rightarrow Y\) with \(T(0)=0\) is linear (where \(X\) and \(Y\) are real normed spaces).NEWLINENEWLINE \textit{D. Tingley} [Geom. Dedicata 22, 371--378 (1987; Zbl 0615.51005)] raised the question whether every surjective isometry between the unit spheres \(S_X\) and \(S_Y\) of two real normed spaces \(X\) and \(Y\) can be extended to a linear isometry between the whole spaces. Tingley himself proved in [loc. cit.] that, if \(X\) and \(Y\) are finite-dimensional, then every surjective isometry \(T:S_X \rightarrow S_Y\) satisfies \(T(-x)=-T(x)\) for every \(x\in S_X\).NEWLINENEWLINE Tingley's isometric extension problem is still open, but there are a number of partial positive results. For example, it is known that Tingley's question has a positive answer in the cases \(X=C(K)\), with \(K\) a compact Hausdorff space, or \(X=L^p(\mu)\), with \(1\leq p\leq\infty\) and \(\mu\) a \(\sigma\)-finite measure.NEWLINENEWLINE In the present paper, the authors use the following terminology in the study of Tingley's problem. A Banach space \(X\) has the Mazur-Ulam property (MUP) if, for every Banach space \(Y\), every surjective isometry \(T:S_X \rightarrow S_Y\) can be extended to a linear isometry between \(X\) and \(Y\).NEWLINENEWLINE \textit{D. N. Tan} and \textit{R. Liu} already proved in [``A note on the Mazur-Ulam property of almost CL-spaces'', J. Math. Anal. Appl. 405, No. 1, 336--341 (2013; \url{doi:10.1016/j.jmaa.2013.03.024})] that every almost CL-space whose unit sphere admits a smooth point has the Mazur-Ulam property (recall that a Banach space \(X\) is called an almost CL-space if the closed absolutely convex hull of every maximally convex subset \(C\) of \(S_X\) equals the whole unit ball).NEWLINENEWLINE In [\textit{K. Boyko} et al., Math. Proc. Camb. Philos. Soc. 142, No. 1, 93--102 (2007; Zbl 1121.47001)], another geometrical notion, the so-called lushness, was introduced in connection with the study of the numerical index of Banach spaces. Every almost CL-space is lush and every lush space has numerical index one (but the converses are both known to be false).NEWLINENEWLINE In the paper under review, the authors introduce the notion of generalized lush (GL) spaces. They prove that every almost CL-space and every separable lush space is GL. Also, they show that \(\mathbb{R}^2\) equipped with the hexagonal norm (which has numerical index \(1/2\) and is therefore in particular not lush) is a GL-space. The main result of the paper is that every GL-space has the MUP.NEWLINENEWLINE A further generalisation of the notion of GL-spaces, called local GL-spaces, is also introduced. The authors show that every lush space is a local GL-space and that local GL-spaces still have the MUP.NEWLINENEWLINE Some stability results are also established. For example, the authors show that the \(c_0\)-, \(\ell^1\)- and \(\ell^{\infty}\)-sum of any family of GL-spaces/local GL-spaces are again GL-spaces/local GL-spaces and that the space \(C(K,X)\) of continuous \(X\)-valued functions on a compact Hausdorff space \(K\) is a GL-space/local GL-space whenever \(X\) is a GL-space/local GL-space.NEWLINENEWLINE It is further proved that, if the \(c_0\)-, resp. \(\ell^1\)-, resp. \(\ell^{\infty}\)-sum of a family \((X_i)_{i\in I}\) of Banach spaces is a GL-space, then each \(X_i\) must be a GL-space. However, the proof of this statement in the \(\ell^1\)-case is not completely correct. An indication of the problem and an alternative proof may be found in the reviewer's preprint [\textit{J.-D. Hardtke}, ``Some remarks on generalised lush spaces'', Preprint (2013), \url{arXiv:1309.4358}, Proposition 2.5].NEWLINENEWLINE At the end of the paper, the following interesting but still open problem is posed by the authors: Does every Banach space with numerical index one have the Mazur-Ulam property?