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 Izv. Akad. Nauk SSSR Ser. Mat., 1983, Volume 47, Issue 1, Pages 3–25 (Mi izv1379)

Integration on Jordan algebras

Sh. A. Ayupov

Abstract: In this paper the author gives a onassociative generalization of abstract integration on JBW-algebras – Jordan Banach algebras having a predual space. Using a faithful normal finite trace on a JBW-algebra $A$, a opology of convergence in measure is introduced and the Jordan algebra $\widehat A$ of all measurable elements with respect to $A$ is constructed as the completion of $A$ in this topology. The spaces $L_1$ and $L_2$ are introduced for $A$ and it is shown that they can be considered as the spaces of all integrable and square-integrable elements, respectively, of $\widehat A$. As in the case of von Neumann algebras it is proved that $L_1$ is isometrically isomorphic to the Banach space predual to $A$.
Bibliography: 33 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1984, 22:1, 1–21

Bibliographic databases:

UDC: 517.986+517.987
MSC: Primary 17C50, 46H99, 46L05; Secondary 12J19, 46A40, 46G12

Citation: Sh. A. Ayupov, “Integration on Jordan algebras”, Izv. Akad. Nauk SSSR Ser. Mat., 47:1 (1983), 3–25; Math. USSR-Izv., 22:1 (1984), 1–21

Citation in format AMSBIB
\Bibitem{Ayu83} \by Sh.~A.~Ayupov \paper Integration on Jordan algebras \jour Izv. Akad. Nauk SSSR Ser. Mat. \yr 1983 \vol 47 \issue 1 \pages 3--25 \mathnet{http://mi.mathnet.ru/izv1379} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=688916} \zmath{https://zbmath.org/?q=an:0532.46038} \transl \jour Math. USSR-Izv. \yr 1984 \vol 22 \issue 1 \pages 1--21 \crossref{https://doi.org/10.1070/IM1984v022n01ABEH001431} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Sh. A. Ayupov, “Locally measurable operators for $JW$-algebras and the representation of ordered Jordan algebras”, Math. USSR-Izv., 24:2 (1985), 199–220
2. N. V. Trunov, “Integration with respect to weight on $JBW$-algebras”, Funct. Anal. Appl., 19:3 (1985), 228–230
3. I. G. Ganiev, A. K. Karimov, “Svoistva skhodimosti po mere na iordanovykh algebrakh”, Vladikavk. matem. zhurn., 5:4 (2003), 43–49
4. M. A. Berdikulov, “Order-unit spaces which are Banach dual spaces”, Zhurn. matem. fiz., anal., geom., 2:2 (2006), 130–137
5. M. A. Berdikulov, “Martingale ergodic processes in Jordan Banach algebras”, Siberian Adv. Math., 23:3 (2013), 175–179
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