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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is isomorphicto 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 at 2:15