$\newcommand\CAR{\mathit{CAR}}\newcommand\Cl{\mathbb C\mathit l}$This question will be rather long and it will be my attempt to finally clarify many issues concerning CCR, CAR and Clifford algebras together with the Fock spaces and the classification of the corresponding von Neumann algebras appearing in this way. I must say that I'm very confused by all this stuff. Let me summarize what I have learned so far. Let $V$ be a Hilbert space (over $\mathbb{R}$ I suppose—at least for the Clifford algebra):

- First of all, let me emphasize the slight difference between the so called CAR (canonical anticommutation relation) algebra over a Hilbert space and the Clifford algebra (in the $C^*$-algebraic version, to be denoted by $\Cl^*(V)$: these two are very closely related. However, note that the Clifford algebra is generated by all $j(v)$'s where $j:V \to \Cl^*(V)$ is the natural embedding (in other words this algebra is the universal $C^*$-algebra satisfying Clifford relations) while the CAR algebra is generated by all creation and anihilation operators (abstractly) so is generated by a priori larger set of relations.

Question 1Is it true that abstractly $\CAR(V)$ and $\Cl^*(V)$ are isomorphic as $C^*$-algebras?

- Now it is shown in Plymen's book „Spinors in Hilbert space‟ that given the von Neumann version of the Clifford algebra (over infinite dimensional $V$) acting via the left regular representation (meaning the GNS representation coming from the trace) one obtains the hyperfinite $II_1$ factor. This $II_1$ hyperfinite factor is unique (up to isomorphism) and admits various incarnations: one of them is an infinite tensor product of $M_2(\mathbb{C})$. This means that $\operatorname{v.N.}\Cl(V)$ acts naturally on the infinite tensor product.
- On the other hand, there are plenty of nonequivalent representations of this Clifford algebra, constructed with the help of the Fock space. The construction of this Fock space requires the choice of the so called unitary (or complex) structure $J$ on our
*real*Hilbert space $V$. Different choices of $J$*can*lead to inequivalent representations and the condition for equivalence is the content of Shale–Stinespring theorem and is of the form that $J-K$ should be a Hilbert–Schmidt operator. - One shows that Fock representations are irreducible: therefore (the v.N. version of) our Clifford algebra will be type $I_{\infty}$. However, we also stated that in the left regular representation our Clifford algebra is just the infinite tensor product $\bigotimes M_2(\mathbb{C})$ which is of type $II_1$. Therefore a natural question arises:

Question 2a) Is there a way to find a representation $\pi$ of $\Cl^*(V)$ such that $(\pi(\Cl^*(V)))''$ is of type $III$? Is the Hilbert space of this representation some kind of infinite tensor product (or wedge product)?

b) Is there a some sort ofcanonicalisomorphism between Fock space and the infinite tensor product (possibly depending whether we take full, or symmetric or antisymmetric Fock space) allowing us to see Fock representation as a representation on this infinite symmetric tensor product?

- Recall that the Fock representation $\pi$ is defined as $\pi(v)=a^+(v)+a^-(v)$ and turns out to be irreducible: thus the von Neumann algebra generated by $\{\pi(v), v \in V\}$ will be the same as the von Neumann algebra generated by $\{a^+(v),a^-(v): v \in V \}$ both being type $I_{\infty}$. Note that the family $\{\pi(v): v \in V\}$ consists from self adjoint operators but the operators $\pi(v)$ do not commute with each other—unlike in the bosonic case for which I would like to turn now.
- So in the bosonic case the analogue of the
*algebraic*version of Clifford algebra is the so called Weyl algebra—however one can show that it is impossible to represent a Weyl algebra as bounded operators in some Hilbert space (unlike for Clifford algebra)—therefore one considers the associated canonical commutation relation in the*integrated*form. One then considers the*universal*$C^*$-algebra generated by these relations. Recall that for the Clifford algebra we had a particular way to represent it as a type $II_1$ hyperfinite factor (using the trace).

Question 3a) Is there a canonical „left regular representation‟ picture for CCR? If the answer is yes, what is the corresponding von Neumann algebra (its type? Is it a factor)?

b) Is there a natural and canonical way to represent CCR algebra as an infinite tensor product?

- For CCR one can again speak about Fock representations and this representation in the bosonic case turns out to be again irreducible: therefore the von Neumann algebra in this representation would be again of type $I_{\infty}$. But note the subtle difference here: the family of field operators $\pi(v):=a^+(v)+a^{-}(v)$ while unbounded, still consists from self adjoint operators but this time this family is
*commutative*! Therefore it should produce some measure: I suspect that this family should produce some variant of Gaussian measure but still I would like to know the details:

Question 4a) If we consider a von Neumann algebra generated by the family $\{\pi(v): v \in V\}$ it should be naturally isomorphic to $L^{\infty}(X,\mu)$. Can one identify the space $(X,\mu)$ concretely?

b) is there a way to arrive at some canonically defined measure starting from CAR instead of CCR? If yes, what is the corresponding measure space? (Some relation with the Ising model?)

- Finally there is also a
*free*(sometimes called*full*) version of the Fock space where one does not use any symmetrization and antisymmetrization. Here the creation and anihilation operators satisfy the following commutation relation: $a^-(u)a^+(v)=\langle u,v \rangle$. Quite surprisingly (at least for me) Voiculescu in his paper „Symmetries of some reduced free product of $C^*$-algebras‟ has proved that the von Neumann algebra generated by $\{a^+(v)+a^-(v): v \in V\}$ in the Fock representation is isomorphic with the von Neumann algebra generated by the free group where the numbers of generators is equal to $\dim V$. So let me ask a final question:

Question 5What von Neumann algebra we get by taking the von Neumann algebra $\{a^+(v),a^-(v): v \in V\}''$ generated by all creation and anihilation operators in thefreeFock space?