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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