Kusraev, A. G. Functional realization of \(AW^*\)-algebras of type I. (English. Russian original) Zbl 0759.46052 Sib. Math. J. 32, No. 3, 416-424 (1991); translation from Sib. Mat. Zh. 32, No. 3(187), 78-88 (1991). In the work of M. Ozawa [J. Math. Soc. Jap. 36, No. 4, 589-608 (1984; Zbl 0599.46083)] the invariants are described that characterize an \(AW^*\)-algebra of type I up to a \(*\)-isomorphism. These invariants are objects of Boolean valued inverse \(V^{(B)}\).In the work being reviewed the analogous invariants are given that do not use the construction of \(V^{(B)}\). Moreover, the representations of \(AW^*\)-modules and \(AW^*\)-algebras of type I are given as spaces of continuous vector-valued functions and as algebras of strongly continuous operator-valued functions, respectively. Reviewer: V.Chilin (Tashkent) Cited in 1 ReviewCited in 1 Document MSC: 46L10 General theory of von Neumann algebras 46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX) 46L05 General theory of \(C^*\)-algebras 03C90 Nonclassical models (Boolean-valued, sheaf, etc.) Keywords:\(AW^*\)-algebra of type I; \(AW^*\)-modules; \(AW^*\)-algebras; spaces of continuous vector-valued functions; algebras of strongly continuous operator-valued functions Citations:Zbl 0599.46083 PDFBibTeX XMLCite \textit{A. G. Kusraev}, Sib. Math. J. 32, No. 3, 416--424 (1991; Zbl 0759.46052); translation from Sib. Mat. Zh. 32, No. 3(187), 78--88 (1991) Full Text: DOI References: [1] G. Takeuti, ?Von Neumann algebras and Boolean-valued analysis,? J. Math. Soc. Jpn.,35, No. 1, 1-21 (1983). · Zbl 0503.46042 · doi:10.2969/jmsj/03510001 [2] G. Takeuti, ?C*-algebras and Boolean-valued analysis,? Jpn. J. Math.,9, No. 2, 207-245 (1983). · Zbl 0538.46060 [3] M. Ozawa, ?A classification of type I AW*-algebras and Boolean-valued analysis,? J. Math. Soc. Jpn.,36, No. 4, 589-608 (1984). · Zbl 0599.46083 · doi:10.2969/jmsj/03640589 [4] M. Ozawa, ?A transfer principle from von Neumann algebras to AW*-algebras,? J. London Math. Soc.,32, No. 2, 141-148 (1985). · Zbl 0626.46052 · doi:10.1112/jlms/s2-32.1.141 [5] M. Ozawa, ?Boolean-valued analysis approach to the trace problem of AW*-algebras? J. London Math. Soc.,33, No. 2, 347-354 (1986). · Zbl 0636.46054 · doi:10.1112/jlms/s2-33.2.347 [6] I. Kaplansky, ?Modules over operator algebras,? Am. J. Math.,75, No. 4, 839-858 (1953). · Zbl 0051.09101 · doi:10.2307/2372552 [7] G. Takeuti and W. M. Zarring, Axiomatic Set Theory, Springer, New York (1973). [8] A. G. Kusraev Vector Duality and Its Applications [in Russian], Nauka, Novosibirsk (1985). [9] A. G. Kusraev, Elements of Boolean-Valued Analysis [in Russian], Mathematics Institute, Siberian Branch, Academy of Sciences of the USSR, Novosibirsk (1987). · Zbl 0639.47023 [10] A. G. Kusraev and S. S. Kutateladze, Nonstandard Methods of Analysis [in Russian], Nauka, Novosibirsk (1980). · Zbl 0448.90062 [11] E. I. Gordon, ?Real numbers in Boolean-valued models of set theory and K-space,? Dokl. Akad. Nauk SSSR,237, No. 4, 773-775 (1977). [12] A. G. Kusraev, ?Banach-Kantorovich spaces,? Sib. Mat. Zh.,26, No. 1, 119-126 (1985). · Zbl 0576.46010 [13] A. G. Kusreev ?To a geometry of Banach spaces with a mixed norm,? Dokl. Akad. Nauk SSSR,300, No. 5, 1049-1052 (1988). [14] A. G. Kusraev, ?Linear operators in lattice normed spaces,? in: Investigations in Geometry ?in the large? and Mathematical Analysis [in Russian], Nauka, Novosibirsk (1987), pp. 84-123. [15] V. A. Lyubetskii and E. I. Gordon, ?Boolean extensions of uniform structures,? in: Investigations in Nonclassical Logics and Formal Systems [in Russian], Nauka, Moscow (1983), pp. 82-153. [16] T. J. Jech, Lectures in Set Theory with Particular Emphasis on the Method of Forcing, Lect. Notes Math.,217 (1971). · Zbl 0236.02048 [17] N. Dunford and J. T. Schwartz, Linear Operators, Interscience, New York (1958-71). This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.