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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.

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