Injective envelopes of real C*- and AW*-algebras

TL;DR

This paper investigates the structure of injective envelopes of real C*-algebras, showing how automorphisms extend and characterizing the nature of their injective envelopes, including examples of type III factors.

Contribution

It establishes the unique extension of outer *-automorphisms to injective envelopes and characterizes the simplicity and type of these envelopes for real C*-algebras.

Findings

Outer *-automorphisms extend uniquely to injective envelopes.

Injective envelopes of simple real C*-algebras are real AW*-factors.

Existence of real C*-algebras with injective envelopes of type III not being W*-algebras.

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.