Integration on Jordan algebras

Sh. A. Ayupov

Izvestiya. Mathematics , Tome 22 (1984) no. 1, pp. 1-21.

Résumé

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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