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arXiv · 2303.05209

Geometry of unit balls of free Banach lattices, and its applications

Timur Oikhberg
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Abstract

We begin by describing the unit ball of the free $p$-convex Banach lattice over a Banach space $E$ (denoted by ${\mathrm{FBL}}^{(p)}[E]$) as a closed solid convex hull of an appropriate set. Based on it, we show that, if a Banach space $E$ has the $\lambda$-Approximation Property, then ${\mathrm{FBL}}^{(p)}[E]$ has the $\lambda$-Positive Approximation Property. Further, we show that operators $u \in B(E,F)$ (where $E$ and $F$ are Banach spaces) which extend to lattice homomorphisms from ${\mathrm{FBL}}^{(q)}[E]$ to ${\mathrm{FBL}}^{(p)}[F]$ are precisely those whose adjoints are $(q,p)$-mixing. Related results are also obtained for free lattices with an upper $p$-estimate.

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Timur Oikhberg. 2023-03-09. Geometry of unit balls of free Banach lattices, and its applications. https://arxiv.org/abs/2303.05209

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