Geometry of unit balls of free Banach lattices, and its applications
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866929220596269056 |
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| author | Oikhberg, Timur |
| author_facet | Oikhberg, Timur |
| contents | 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 $λ$-Approximation Property, then ${\mathrm{FBL}}^{(p)}[E]$ has the $λ$-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_05209 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Geometry of unit balls of free Banach lattices, and its applications Oikhberg, Timur Functional Analysis 46B42, 46B28, 47B10 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 $λ$-Approximation Property, then ${\mathrm{FBL}}^{(p)}[E]$ has the $λ$-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. |
| title | Geometry of unit balls of free Banach lattices, and its applications |
| topic | Functional Analysis 46B42, 46B28, 47B10 |
| url | https://arxiv.org/abs/2303.05209 |