Description of Real AW*- Factors of Type I

Abstract

The paper of the is to initiate the study of real AW*-algebras in the framework of the theory of real C*-algebras and W*-algebras. It happens that in some aspects real AW*-algebras behave unlike complex AW*-algebras and sometimes their properties are completely different also from corresponding properties of real W*-algebras. We prove that if the complexification  of a real C*-algebra A is a (complex) AW*-algebra then A itself is a real AW∗-algebra. By modifying the Takenouchi’s examples of complex non-W*, AW*-factors we show that there exist real non-W*, AW*-factors. The correspondence between real AW*-factors and involutive (i.e. with period 2) *-anti-automorphisms of (complex) AW*-factors is established. We give the decomposition of real AW*-algebras into types I, II and III similar to the case of complex AW*-algebras or W*-algebras. It is proved that if A is a real AW*-factor and its complexification  is also an AW*-algebra (and therefore an AW*-factor) thenthetypesof A and M coincide.