This question is from the Sunder's book An Invitation to von Neumann Algebras Ex 4.2.13

Let M be a semifinite factor with fns trace ${\tau}$. Let ${\theta}$ be an automorphism of M such that ${\tau}$ ${\theta}$ = ${\lambda}$ ${\tau}$ for some ${\lambda}$ in (0,1). Define ab action ${\alpha}$ of Z on M by ${\alpha}$$_n$ = ${\theta}$$^n$, prove that the crossed product $\widetilde M%$ is of type III$_{\lambda}$.

There is a hint to used the following prop 4.2.9, but I also cannot catch the ideal of prop 4.2.9, it seems that the condition (ii) is complicated, please help.

enter image description here


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.