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Octahedral norms in free Banach lattices. (English) Zbl 1458.46009
The free Banach lattice generated by just a set was introduced by B. de Pagter and A. W. Wickstead [Proc. R. Soc. Edinb., Sect. A, Math. 145, No. 1, 105–143 (2015; Zbl 1325.46020)]. A couple of years later, free Banach lattices generated by Banach spaces were introduced by A. Avilés et al. [J. Funct. Anal. 274, No. 10, 2955–2977 (2018; Zbl 1400.46015)].
Let \(X\) be a Banach space and denote by \(FBL[X]\) the free Banach lattice generated by \(X\). In the paper under review, the authors investigate sufficient conditions on \(X\) to assure that the norm of \(FBL[X]\) is octahedral.
Recall that a Banach space \(X\) has an octahedral norm if, for every finite-dimensional subspace \(E\) of \(X\) and for every \(\varepsilon>0\), there is a \(y\in X\) with norm one such that \[ \|x+\lambda y\|\geq (1-\varepsilon)(\|x\|+|\lambda|) \] for every \(x\in E\) and \(\lambda\in \mathbb{R}\).
The main results of the paper say that the \(FBL[X]\) is octahedral whenever the Cunningham algebra \(C(X^{(\infty})\) is infinite-dimensional (e.g., \(X\) is \(L\)-embedded) or \(X^{\ast}\) is almost square. The paper ends by pointing out a couple of interesting open questions.

MSC:
46B04 Isometric theory of Banach spaces
46B20 Geometry and structure of normed linear spaces
46B42 Banach lattices
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