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Boolean Valued Analysis of Banach Spaces

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Abstract

We implement the Boolean valued analysis of Banach spaces. The realizations of Banach spaces in a Boolean valued universe are lattice normed spaces. We present the basic techniques of studying these objects as well as the Boolean valued approach to injective Banach lattices.

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Notes

  1. This means that the underlying ring of the poset \( \operatorname{Orth}^{\infty}(E) \) is a lattice and multiplication by every positive members of \( E \) is a positive band preserving operator; see [8, Definition 2.53].

  2. In the terminology of Russian provenance, the shorter term foundation is used instead of order dense ideal, which explains the above definition.

  3. Also known as Escher rules.

  4. An element \( v \) in a vector lattice \( X \) is a component or fragment of \( u\in X \) if \( |v|\wedge|u-v|=0 \).

  5. In case \( \|T\|\leq\lambda\|T_{0}\| \) with \( 1\leq\lambda\in{𝕉} \) replaces \( \|T_{0}\|=\|T\| \) in the definition of injective Banach lattice, the terms like \( \lambda \)-injectivity are applied. In this article we confine exposition to the isometric theory; i.e., we consider only 1-injective Banach lattices.

  6. The zero Banach lattice \( X=\{0\} \) is injective formally, but \( X \) fails to satisfy the claims of Corollary 7.11(1); since the Boolean algebra \( B=M(X)=\{0\} \) and the unverse \( {𝕍}^{(B)} \) degenerate in this case. By default we assume \( X \) nonzero in all similar situations.

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Funding

The research of Kusraev was executed in the North Caucasus Mathematical Research Center of the Vladikavkaz Scientific Center of the Russian Academy of Sciences and supported by the Ministry of Science and Higher Education of the Russian Federation (Agreement 075–02–2023–914). The research of Kutateladze was carried out in the framework of the State Task to the Sobolev Institute of Mathematics (Project FWNF–2022–0004).

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Authors and Affiliations

  1. Southern Mathematical Institute, Vladikavkaz, Russia

    A. G. Kusraev

  2. North Caucasus Center for Mathematical Research, Mikhailovskoye Village, Russia

    A. G. Kusraev

  3. Sobolev Institute of Mathematics, Novosibirsk, Russia

    S. S. Kutateladze

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  1. A. G. Kusraev
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  2. S. S. Kutateladze
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Correspondence to A. G. Kusraev.

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Kusraev, A.G., Kutateladze, S.S. Boolean Valued Analysis of Banach Spaces. Sib Math J 65, 190–233 (2024). https://doi.org/10.1134/S0037446624010178

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