Stably Properly Infinite and Purely Infinite Real C*-algebras

This paper considers real C*- and AW*-algebras with quasitrace and explores their connection with the concept of stable finiteness, stable proper infiniteness and pure infiniteness. A review of existing results for complex algebras and their real analogues is presented, including the role of quasitrace in concept of finiteness and infiniteness. It is proved that a real C*-algebra is stably properly infinite if and only if its complexification has the same property. The concept of quasitrace allows us to classify C*-algebras as finite or infinite. If the quasitrace is trivial, such an algebra is called traceless or stably properly infinite. There are known results such as a connection between traceless and weakly purely infinite complex C*-algebras and equivalence of all definitions of pure infiniteness under condition of real rank zero, so a natural question arises: how are traceless C*-algebras related to infinite ones in the real case and if all definitions of pure infiniteness under condition of real rank zero coincide in real case? In this paper, we obtain the following results: a real analogue of the Cuntz-Blackadar-Handelman theorem, build the connection between weakly purely infinite and traceless real C*-algebras through their enveloping complex C*-algebras, and we give Stacey's theorem without the assumption simplicity through the class of stably properly infinite (traceless) real C*-algebras. It would be reasonable to consider AW*-algebras, since we consider real rank zero C*-algebras as the class AW*-algebras lies in the class of real rank zero C*-algebras.