Von Neumann algebra associated to the infinite Cuntz algebra

The von Neumann algebra $M$ generated by $\mathcal O_\infty$ is all of $B(\mathcal F(H))$.

Indeed, if $a$ belongs to its commutant, let me prove that $a$ is a multiple of the identity. First since for all $v \in H$, $s_v^* (a \Omega)= a (s_v^* \Omega)=0$, we have that $a \Omega=\lambda \Omega$ for some $\lambda \in \mathbb C$. Then for every $\xi \in \oplus_n H^{\otimes n}$ (finite sum) pick $x \in \mathcal O_\infty$ such that $x \Omega=\xi$. Then $a(\xi) = x (a\Omega)=\lambda \xi$. This proves $a=\lambda$.

(Let me add that, according to the standard notation, $\Omega$ here denotes some fixed unit vector in $H^{\otimes 0}$).

I'm reasonably convinced the algebra generates all of $B(\mathcal{F}(H))$. Note that $1-\sum_{i=1}^\infty s_is_i^*$ converges strongly to the projection $P\_0$ onto $\mathbb{C}$ and similarly the sum $1-\sum_{|\alpha|=n}s_\alpha s_\alpha^*$ over multi-indices converges to the projection $P_n$ onto vectors with highest tensor power $n-1$. Pick vectors $u\in H^{\otimes n}$, $v\in H^{\otimes m}$, then the operators $s_u$, $s_v$ (the obvious generalisations of the $s_i$) are in your algebra and so is the rank one operator $s_vs_u^*P_{n+1}=|v><\ u|$. Finally note you may approximate any rank one in $B(\mathcal{F}(H))$ by these.

If you take the Fock space $\mathcal F_P$ associated to a polarization of $H$, as in [W, page 480, line 7], and consider the CAR algebra (=algebra generated by creation and annihilation operators) generated by a subspace $K\subset H$, as in [W, page 497, line 29], then the von Neumann algebra this generates is (hyperfinite) of type $III_1$ [W, Corollary on the bottom of page 500].

Reference:
[W] Wassermann, Operator algebras and conformal field theory