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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
##### Citations:
Zbl 1325.46020; Zbl 1400.46015
Full Text:
##### References:
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