I should be tarred and feathered for not knowing at least the status of the following question. 



> 
> **Question:** Let $\Gamma$ be a discrete amenable group. If $\pi:\Gamma \rightarrow B(\mathcal{H})$ is a unitary representation of $\Gamma$ on a separable Hilbert space $\mathcal{H}$, is the von Neumann algebra $\pi(\Gamma)''$ necessarily injective?
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> 


Flippantly one imagines that the answer to this question is yes, by Theorem 2.2 of [Bekka's paper](https://docs.google.com/viewer?a=v&pid=gmail&attid=0.1&thid=13b47826ab93ad07&mt=application/pdf&url=https%3A//mail.google.com/mail/u/0/?ui%253D2%2526ik%253D37934658d5%2526view%253Datt%2526th%253D13b47826ab93ad07%2526attid%253D0.1%2526disp%253Dsafe%2526realattid%253Df_ha2kpjyg0%2526zw&sig=AHIEtbSG9so6BO0JmfYXIVDarAqT6atkWA) on amenable representations. But this result only says that the images of group elements are in the centralizer of a non-normal state...it isn't immediately clear why the entire von Neumann algebra should lie in the centralizer of such a state. If one tries to sidestep this by looking at a proof using almost invariant vectors, one is busted by the fact that a representation that is "[$H$-amenable](http://www.mscand.dk/article.php?id=2922)" isn't necessarily amenable in Bekka's sense. 


**EDIT:** Makoto's nice answer below provided me with some closure. I'm still worried that I can't see a more or less direct way to this result from Connes's '76 paper on the classification of injective factors. If this paper can, in a more or less direct and self-contained way, be used to resolve the question, please feel free to include another answer.



Is the von Neumann algebra associated to a unitary representation of an amenable group always injective?
This is because you get a representation of the maximal(=reduced by amenability) C\*-algebra $C^\ast(\Gamma)$ on $H$, and the image has to be a nuclear algebra because it's a quotient of nuclear one. Then, $\pi(\Gamma)''$, being a weak\*-closure of a nuclear C\*-algebra, has to be injective.


