Structure of Injective Real Factors of Type III${\text{III}}_\lambda ,0 < \lambda < 1$

Abstract

We consider ^*-automorphisms and ^*-antiautomorphisms of real and complex factors. We establish both the uniqueness of the class of ^*-automorphisms (with $\bmod \left( \right) = \lambda ,\lambda \ne 1$) of a real injective ${\text{II}}_\infty$ factor and the uniqueness of the class of ^*-antiautomorphisms (with $\bmod \left( \right) = \sqrt \lambda ,\lambda \ne$) of a complex injective ${\text{II}}_\infty$ factor. It is well known that, for complex factors, the notions of hyperfiniteness and injectivity are equivalent. Here we prove that, for real factors, the two notions are no longer equivalent. Bibliography: 15 titles.