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Ordered algebras. (Uporyadochennye algebry). (Russian) Zbl 0542.46001
Akademiya Nauk Uzbekskoj SSR. Institut Matematiki Imeni V. I. Romanovskogo. Tashkent: Izdatel’stvo ”Fan” Uzbekskoj SSR. 304 p. R. 3.10 (1983).
This book is devoted to a detailed study of two classes of ordered algebras, mamely, the OJ-algebras and the \(O^*\)-algebras. A JBW-algebra is a typical example of an OJ-algebra and, a von Neumann algebra or the algebra of measurable operators associated with a von Neumann algebra is an \(O^*\)-algebra. These algebras have been used in quantum mechanical formalism and noncommutative probability theory and indeed, as the authors explain, they arose from an attempt to give an algebraic description of the spaces of random variables in classical as well as noncommutative probability theory.
An OJ-algebra A is a unital real Jordan algbra equipped with a partial ordering which is boundedly complete and is such that every maximal strongly associative subalgebra is a lattice. The subalgebra of all bounded elements in A form a JB-algebra and the set of all idempotents in A form a complete logic.
An \(O^*\)-algebra can be regarded as a ”complexification” of an OJ- algebra, it is a partially ordered (associative) complex \({}^*\)-algebra which is boundly complete and also the self-adjoint part of each maximal commutative \({}^*\)-subalgebra is a lattice. Thus the self-adjoint part of an \(O^*\)-algebra is an OJ-algebra.
The book consists of five chapters and the titles are: 1. Logics; 2. Semifields; 3. Ordered Jordan algebras; 4. Ordered algebras with an involution; 5. Topological \(O^*\)-algebras.
Reviewer: Chu Cho-Ho

MSC:
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
46A40 Ordered topological linear spaces, vector lattices
46L99 Selfadjoint operator algebras (\(C^*\)-algebras, von Neumann (\(W^*\)-) algebras, etc.)
17C65 Jordan structures on Banach spaces and algebras
46L51 Noncommutative measure and integration
46L53 Noncommutative probability and statistics
46L54 Free probability and free operator algebras
06F25 Ordered rings, algebras, modules
47A40 Scattering theory of linear operators
46K05 General theory of topological algebras with involution