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The Cuntz algebra $\mathcal{O}_{\infty}$ is the universal $C^*$-algebra generated by countably infinitely many isometries $s_i$ satisfying the relations $s_i^*s_j = \delta_{ij}$ (there is no condition about the sum $\sum_i s_is_i^*$ in this case, since the sum would be infinite).

There is another way to describe this $C^*$-algebra as a subalgebra of the bounded operators on a Hilbert space as follows: Let $H$ be a separable Hilbert space of infinite dimension. Then we define the Fock space

$$ \mathcal{F}(H) = \bigoplus_{n \in \mathbb{N}_0} H^{\otimes n}\ . $$

with $H^{\otimes 0} \cong \mathbb{C}$. Any element $v \in H$ defines a creation operator $s_v(\xi) = v \otimes \xi$ and an annihilation operator $s_v^*(w \otimes \xi) = \langle v, w \rangle \xi$, which is zero on $\mathbb{C} = H^{\otimes 0} \subset \mathcal{F}(H)$. The $C^*$-algebra generated by $s_v$ and $s_v^*$ (i.e. the norm closure) is again $\mathcal{O}_{\infty}$. After choosing an orthonormal basis $e_i \in H$, we can identify $s_i = s_{e_i}$.

My question is:

Is there anything known about the von Neumann algebra generated by $\mathcal{O}_{\infty}$ in this way, i.e. the weak closure or double commutant of $\mathcal{O}_{\infty}$ in $B(\mathcal{F}(H))$? What type is it ($III_1$ or just $I_{\infty}$)?

EDIT: Are there interesting/canonical states on $\mathcal{O}_{\infty}$ such that the von Neumann algebra associated to the GNS-construction is type $III_1$?

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up vote 17 down vote accepted

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 $H^{\otimes 0}$).

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Snap! Nice proof. –  Ollie Margetts Aug 30 '12 at 12:06
    
That is nice. Thank you! –  Ulrich Pennig Aug 30 '12 at 13:01
    
Do you know about any states on $\mathcal{O}_{\infty}$ which would yield a type $III$-von Neumann algebra? –  Ulrich Pennig Aug 30 '12 at 13:45
    
I do not know, but I am not very familiar with $\mathcal O_\infty$ (and with $C^*$-algebras in general). Do you have an argument why such a state should exist? –  Mikael de la Salle Aug 30 '12 at 18:33
    
What about these free quasi-free states: msp.berkeley.edu/pjm/1997/177-2/pjm-v177-n2-p07-p.pdf? (I haven't looked carefully, this is an off the cuff comment...) –  Jon Bannon Aug 30 '12 at 18:41
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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.

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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

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This is one of the reasons, I was interested in the question. The Cuntz algebra $\mathcal{O}_{\infty}$ has some properties in common with the ($C^*$-algebraic) CAR-algebra - for example both are strongly self-absorbing. But it is purely infinite and therefore contains no finite projection, which in my head "moves it closer to" type $III_1$-factors. –  Ulrich Pennig Jul 29 '13 at 8:47
    
I'm confused by your remark: didn't you say that $\mathcal O_\infty$ is isomorphic to the $C^*$-algebraic CAR algebra? Now you seem to indicate that they are only similar? –  André Henriques Jul 29 '13 at 11:04
    
No, that is not what I said. The CAR-algebra is isomorphic to the UHF-algebra $M_{2^{\infty}}$ (see Example 1.2.6 in Rordam's "Classification of Nuclear, Simple $C^*$-algebras"). I might have said that the methods in our paper apply to the CAR-algebra as well, which is true since both are strongly self-absorbing. –  Ulrich Pennig Jul 29 '13 at 12:14
    
What might have caused the confusion is that the descriptions in terms of faithful representations on Fock space really look very similar, but note that for $\mathcal{O}_{\infty}$ I need to take the full tensor algebra and not just the antisymmetric part. If I only take the latter and take creation and annihilation ops then I get CAR. –  Ulrich Pennig Jul 29 '13 at 12:19
    
I see. Thanks for the clarification. –  André Henriques Jul 29 '13 at 14:08
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