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Boolean Valued Analysis: Background and Results

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Operator Theory and Differential Equations

Part of the book series: Trends in Mathematics ((TM))

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Abstract

The paper provides a brief overview of the origins, methods and results of Boolean valued analysis. Boolean valued representations of some mathematical structures and mappings are given in tabular form. A list of some problems arising outside the theory of Boolean valued models, but solved using the Boolean valued approach, is given. The relationship between the Kantorovich’s heuristic principle and the Boolean valued transfer principle is also discussed.

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Notes

  1. 1.

    A K-space or a Kantorovich space is a Dedekind complete vector lattice. An extended K-space is a universally complete vector lattice, cp. [4] and [104].

  2. 2.

    The base of a vector lattice is the inclusion ordered set of all of its bands (that forms a complete Boolean algebra) [36, 104].

  3. 3.

    Earlier G. Saks [88] without assumption of existence of inaccessible cardinal proved that the statement “The Lebesgue measure on \(\mathbb {R}\) can be extended to the σ-additive invariant measure defined on all subsets of \(\mathbb {R}\)” is consistent with ZF + DC.

  4. 4.

    H: Every order complete order dense linearly ordered set having no first or last element is order isomorphic to the ordered set of reals \(\mathbb {R}\), provided that every collection of mutually disjoint non-empty open intervals in it is countable.

  5. 5.

    NDH: For each compact space X, each homomorphism from \(C(X,\mathbb {C})\) into a Banach algebra is continuous.

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Authors and Affiliations

  1. Southern Mathematical Institute, Vladikavkaz Scientific Center of the RAS, Vladikavkaz, Russia

    A. G. Kusraev

  2. Sobolev Institute of Mathematics, Siberian Branch of the RAS, 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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  1. Southern Mathematical Institute of Vladikavkaz Scientific Center, Russian Academy of Sciences, Vladikavkaz, Russia

    Anatoly G. Kusraev

  2. Southern Mathematical Institute of Vladikavkaz Scientific Center, Russian Academy of Sciences, Vladikavkaz, Russia

    Zhanna D. Totieva

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Kusraev, A.G., Kutateladze, S.S. (2021). Boolean Valued Analysis: Background and Results. In: Kusraev, A.G., Totieva, Z.D. (eds) Operator Theory and Differential Equations. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-49763-7_9

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