Injective Envelopes of Real C*- and AW*-Algebras

Injective (complex and real) W*- and C*- algebras, in particular, factors have been studied quite well. On the other hand, in an arbitrary case, i.e., in the non-injective case, it is quite difficult to study (up to isomorphism) the W*-algebras, in particular, it is known that there is a continuum of pairwise non-isomorphic non-injective factors of type II<sub>1</sub>. Therefore, it seems interesting to study the so called maximal injective W* and C*-subalgebras or what is equivalent, the smallest injective C*-algebra containing a given algebra, which is called an injective envelope of C*- algebra. It is shown that every outer *-automorphism of a real C*-algebra can be uniquely extended to an injective envelope of real C*-algebra. It is proven that if a real C*-algebra is a simple, then its injective envelope is also simple, and it is a real AW*-factor. The example of a real C*-algebra that is not real AW*-algebra and the injective envelope is a real AW*-factor of type III, which is not a real W*-algebra is constructed. This leads to the interesting result that up to isomorphism, the class of injective real (resp. complex) AW*-factors of type III is at least one larger than the class injective real (resp. complex) W*-factors of type III.