# Enveloping von Neumann algebra of Clifford algebra

As explained in the book "Spinors in Hilbert Space" by Plymen and Robinson, if $$V$$ is a complex (separable) Hilbert space with a real structure, and $$\mathrm{Cl}(V)$$ the corresponding Clifford algebra, there is a unique completion $$\mathrm{Cl}[V]$$ of this algebra to a $$C^*$$-algebra: Any $$*$$-representation of $$\mathrm{Cl}(V)$$ on a Hilbert space will induce that same $$C^*$$-norm on it, and the corresponding closure is the Clifford $$C^*$$-algebra $$\mathrm{Cl}[V]$$.

The situation is different for von-Neumann algebras. If $$\pi: \mathrm{Cl}(V) \rightarrow B(\mathcal{H})$$ is a $$*$$-representation, then the von-Neumann-closure $$M_\pi := \pi(\mathrm{Cl}(V))^{\prime\prime}$$ will depend drastically on $$\pi$$. For example, if $$\pi$$ is the left regular representation, then $$M_\pi$$ is the hyperfinite $$\mathrm{II}_1$$ factor, if $$\pi$$ is a Fock representation, then it is type $$\mathrm{I}_\infty$$, and there are other settings where we obtain type $$\mathrm{III}$$ factors.

However, there is another canonical von-Neumann algebra containing $$\mathrm{Cl}(V)$$, namely the universal enveloping algebra, coming from the case where $$\pi$$ is the universal representation of $$\mathrm{Cl}[V]$$. Alternatively, it is the double dual of $$\mathrm{Cl}[V]$$.

Q: What can we say about the enveloping von-Neumann-algebra of $$\mathrm{Cl}[V]$$? Does it happen to be a factor? Is it possibly the hyperfinite $$\mathrm{II}_1$$ factor?

• No way! Every von Neumann completion is a quotient of the universal enveloping algebra. It's as far from being a factor as you can get. – Nik Weaver May 12 at 5:51
• I know that there is a unique map to every von-Neumann completion; but how do you see it is surjective? – Matthias Ludewig May 12 at 12:13
• Any weak* closed ideal of a von Neumann algebra $M$ has the form $pM$ for some central projection $p$. The quotient $M/pM$ is isomorphic to $(1-p)M$. Thus the image of $M$ under any normal (i.e., weak* continuous) $*$-homeomorphism is a von Neumann algebra. – Nik Weaver May 12 at 15:38