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Boolean Valued Analysis of Banach Algebras

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Abstract

We implement the Boolean valued analysis of Baer \( C^{\ast} \)-algebras and Jordan–Banach algebras. These algebras transform into \( AW^{\ast} \)- and \( JB \)-factors. Presentation of the factors as operator algebras leads to Kaplansky–Hilbert modules. We overview the basic properties of these objects.

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Notes

  1. As usual, \( \operatorname{ext}(K) \) is the set of extreme points of a convex set \( K \).

  2. Cp. [1, Theorem 5.2.6(10)].

  3. These are often called arrow cancelation rules; cp. [1, 3.3.12(6)].

  4. The phrase “a projection \( \pi \) contains a projection \( \rho \)” means that \( \rho\leq\pi \).

  5. Cp. [1, 2.5.7] and [12, 5.1].

  6. Cp. [14, Chapter 1].

  7. Cp. [15, p. 38].

  8. Cp. [1, 2.3.3].

  9. Cp. [13, p. 38].

  10. This is defined as \( [\lambda]\!:=\bigwedge\{b\in 𝔹:b\lambda=\lambda\} \) for all \( \lambda\in\Lambda \); cp. [1, 4.2.6].

  11. Cp. [4, vol. 1, p. 329].

  12. Cp. [20, Section 3].

  13. Cp. [14, p. 361].

  14. Recall that \( \mathfrak{P}(M) \) is a Boolean algebra in the commutative case.

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Funding

The research was supported by the Ministry of Science and Higher Education of the Russian Federation (Agreement 075–02–2023–914) 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 S. S. Kutateladze.

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The article was submitted by the authors in English.

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Kusraev, A.G., Kutateladze, S.S. Boolean Valued Analysis of Banach Algebras. Sib Math J 64, 1001–1034 (2023). https://doi.org/10.1134/S0037446623040225

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