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In Connes' book Noncommutative geometry, there is a presentation of all hyperfinite factors. He reffers to type $II_1$ as the Clifford algebra of infinite dimensional Euclidean space. This factor can be constructed as a von Neumann algebra of the i.c.c (discrete) group of all permutations of $\mathbb{Z}$ leaving fixed all integers except of finite number of them. From this construction it is not clear for me what it has to do with Clifford algebra. Could anybody give me a good reason for such a name?

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The hyperfinite $II_1$-factor has many manifestations. Start with the Clifford algebra of a $\mathbb C^n$ and study how it sits inside the Clifford algebra of $\mathbb C^{n+1}$. – Andreas Thom Aug 11 '14 at 18:33
I am far from an expert. My impression is that "the hyperfinite II_1 factor" has lots of automorphisms, and different manifestations of it, although isomorphic, often are not canonically isomorphic. The classification of factors only says that a given one is isomorphic to the hyperfinite II_1 factor, and doesn't tell you that they are really "the same" in any meaningful way. But I repeat: I am far from an expert. – Theo Johnson-Freyd Aug 12 '14 at 2:15

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