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The crossed product of group $\mathbb{R}$ of real numbers with type $II_{\infty}$ is type $III$ factor when the action of $\mathbb{R}$ on von Neumann algebra of type $III$ factor is ergodic and free. This example I know. Can you give some example of type $III$ factors not coming from a crossed product? I want the typical constructive example.

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By Theorem XII.1.1 in Takesaki's Theory of Operator Algebras, every von Neumann algebra of type III is isomorphic to the crossed product of some von Neumann algebra with real numbers, so such an example does not exist.

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