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The monograph is devoted to the real counterpart of the theory of operator algebras, namely to the theory of real W -algebras. We develop a real analogue of index theory for subfactors of W - factors and study actions of various kinds of groups (finite, discrete, amenable, abelian, compact) on different types of real W -algebras. An abstract (non spatial) generalization of real W -algebra - so called real AW -algebras are also investigated. These algebras present a real counterpart of Kaplansky algebras - complex AW -algebras. Type decomposition for real AW -algebras is obtained and relation between types of a real AW -algebra and its enveloping complex AW -algebra is investigated. Unbounded extensions of real W -algebras are introduced in an abstract algebraic manner in terms of real partially ordered involutory algebras - real O -algebras. We prove a representation theorem for abstract real O -algebras as algebras of locally measurable operators affiliated with a real W - algebra. One of the main tools in investigation of the above problems is the reduction to the complex enveloping algebras and their involutive - antiautomorphisms.

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