# Hyperfinite type II_1 factor as the Clifford algebra

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?

• 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

First, note that there are many manifestations of the hyperfinite $II_{1}$ factor, but it turns out that they are all isomorphic.

I will thus interpret your question as asking how the Clifford algebra of infinite-dimensional Euclidean space is a hyperfinite $II_{1}$ factor. The book Spinors in Hilbert space by Plymen & Robinson answers this question in paragraph 1.3.

Of course, the authors of the book give a far better explanation of the story, but let me try to give a sketch anyway.

Let $V$ be a separable infinite dimensional real Hilbert space, endowed with a unitary structure $J:V \rightarrow V$. (That is, an orthogonal map that squares to $-1$).

The complex Clifford algebra $C(V)$ is endowed with a complex anti-linear involution, and can be completed into a $C^{*}$-algebra, called the Clifford $C^{*}$-algebra and denoted $C[V]$. The Clifford algebra is equipped with a canonical trace $\tau: C(V) \rightarrow \mathbb{C}$, and thus with an inner product defined by $\langle \xi | \eta \rangle = \tau(\eta^{*} \xi)$. Completing the space $C(V)$ with respect to this inner product gives us a Hilbert space $\mathbb{H}_{\tau}$. The Clifford algebra $C(V)$ acts via the left regular representation on $\mathbb{H}_{\tau}$, and this representation can be extended to the Clifford $C^{*}$-algebra $C[V]$. Write $\lambda: C[V] \rightarrow B(\mathbb{H}_{\tau})$ for this representation.

The von Neumann Clifford algebra $\mathcal{A}[V]$ is now defined as the weak closure of $\lambda(C[V])$, or equivalently $\mathcal{A}[V] := \lambda(C(V))''$, where a prime denotes the commutant.

Finally, one may show that the von Neumann Clifford algebra $\mathcal{A}[V]$ has a unique central normal faithful tracial[*] state $\tau: \mathcal{A}[V] \rightarrow \mathbb{C}$. One then uses this to show that $\mathcal{A}[V]$ is type $II_{1}$. The fact that it is hyperfinite follows from the fact that $V$ is seperable.

*I am not sure if there is any redundancy between these adjectives.