On the structure of real injective $W^*$-factors of type III$_{\lambda}$, $0<\lambda<1$

We consider $*$-automorphisms and $*$-antiautomorphisms of real and complex factors. We establish both the uniqueness of the class of $*$-automorphisms (with $\mod(\cdot )=\lambda$, $\lambda\ne1$) of a real injective II$_\infty$ factor and the uniqueness of the class of $*$-antiautomorphisms (with$\mod(\cdot )=\sqrt{\lambda}$, $\lambda\ne1$) of a complex injective 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.