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Hi, It is known that the double commutant of the CCR algebra in it's GNS space with respect to some quasi-free states are always type III factors. My question is; Will some of them be hyperfinite factors?

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  • $\begingroup$ Sorry for confusing Ccr with ccr;-) $\endgroup$
    – Marc Palm
    Commented Mar 21, 2013 at 16:18
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    $\begingroup$ Just to pick up on Marc's comment: by CCR you mean canonical commutation relations, not completely continuous representations (aka liminal), right? $\endgroup$
    – Yemon Choi
    Commented Mar 21, 2013 at 18:25
  • $\begingroup$ I answered that the later are alway type 1, but the op referred to the former. $\endgroup$
    – Marc Palm
    Commented Mar 22, 2013 at 14:23

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Yes, Araki-Woods showed they're always ITPFI factors and ITPFI factors are hyperfinite. See the following:

Araki-Woods, A classification of factors

It's a bit of a monster paper, the stuff on CCR algebras is near the end.

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