Injective envelopes of real C*- and AW*-algebras
A. A. Rakhimov, L. D. Ramazonova
TL;DR
The paper addresses the problem of extending outer *-automorphisms from real $C^*$-algebras to their injective envelopes and analyzes the structure of these envelopes, particularly for simple algebras. It employs complexification via $A=R+iR$ and Hamana's injective envelope framework to show that outer automorphisms extend uniquely to the envelope and that, for simple real algebras, the injective envelope is a real $AW^*$-factor. A key contribution is constructing a real example where the injective envelope is a real $AW^*$-factor of type III that is not a real $W^*$-algebra, illustrating that the class of injective real (and complex) type III AW*-factors is strictly larger than their W*- counterparts. This study thus reveals a richer landscape for injective envelopes in the real setting and has implications for the classification of type III injective factors across real and complex cases.
Abstract
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.