Let $H$ be an infinite dimensional separable Hilbert space.
The $C^{*}$-algebras and von Neumann here unital and subalgebras of $B(H)$.
Definition : Let $\mathcal{A}$ be $C^{*}$-algebra (resp. a von Neumann algebra), $\mathfrak{gen}(\mathcal{A})$ is the fewest number of self-adjoint operators generating $\mathcal{A}$ as $C^{*}$-algebra (resp. as von Neumann algebra).
Remark: If $\mathfrak{gen}(\mathcal{A}) \leq 2$ then $\mathcal{A}$ is called singly-generated, because if $x$ and $y$ are self-adjoint generators and $T = x+iy$, then $x = (T+T^{*})/2$ and $y = (T+T^{*})/2i$.
Remark: Let $\mathcal{A}$ be a $C^{*}$-algebra then $\mathfrak{gen}(\mathcal{A''}) \leq \mathfrak{gen}(\mathcal{A})$.
Examples :
$\mathfrak{gen}(C([0,1]^{n}) = n$ and $\mathfrak{gen}(L^{\infty}([0,1]^{n})) = 1$ (because $L^{\infty}([0,1]^{n}) \simeq L^{\infty}([0,1])$)
$\mathfrak{gen}(C^{*}_{r}(\mathbb{F}_{n})) \leq 2n$ because $\{a_{k}+a_{k}^{-1},i(a_{k}-a_{k}^{-1}) \vert k=1...n \}$ generates all.
$\mathfrak{gen}(L(\mathbb{F}_{2})) = 2 $ (singly-generated):
$L(\mathbb{F}_{2}) = W^{*}(a_{1},a_{2})$, however $W^{*}(a_{1}) \simeq L^{\infty}([0,1]) = W^{*}(S)$ with $S$ self-adjoint. Then $\exists S_{1}, S_{2} \in B(H)$ self-adjoint, such that $W^{*}(a_{k}) = W^{*}(S_{k})$. So $L(\mathbb{F}_{2}) = W^{*}(S_{1}+iS_{2})$.Idem: $\mathfrak{gen}(L(\mathbb{F}_{n})) \leq n $
In this paper (example 5 page 4) Masaru Nagisa shows that the full group $C^{*}$-algebra $C^{*}(\mathbb{F}_{2})$ generated by the free group $\mathbb{F}_{2}$, is not singly generated.
Question 1 : Is the problem of whether "$ C^{*}_{r}(\mathbb{F}_{n}) $ is singly generated" still open ?
(Narutaka Ozawa : "yes")
Generator problem for von Neumann algebras : Are
every von Neumann algebras singly generated ?
Examples (see here) : those hyperfinite, those with a Cartan subalgebra, $L(\mathbb{F}_{2})$.
Still open (see here) : the free group factor $L(\mathbb{F}_{n})$, $n>2$.
Remark: $L(\mathbb{F}_{n})=C^{*}_{r}(\mathbb{F}_{n})''$, so $C^{*}_{r}(\mathbb{F}_{n})$ is (still) not known to be singly generated (question 1).
Free group factor isomorphism problem : $L(\mathbb{F}_{n}) \simeq L(\mathbb{F}_{m})$, $\forall n, m \geq 2$ ?
Remark : Pimsner and Voiculescu showed (here 1982, 8(1)) that $C^{*}_{r}(\mathbb{F}_{n}) \not\simeq C^{*}_{r}(\mathbb{F}_{m}) $ by computing their $K$-theory : $ K_{1}(C^{*}_{r}(\mathbb{F}_{n})) = \mathbb{Z}^{n}$.
Question 2 : Are there free interpolated $C^{*}$-algebras $C^{*}_{r}(\mathbb{F}_{s})$ ? $\not\simeq$ ? What about their $K$-theory ?
(N. Ozawa: "I don't know") $\to$ (see question 4)
Theorem (here p 137) : Let $\mathcal{A}$ be $C^{*}$-algebra and $r = \mathfrak{gen}(\mathcal{A}) $ then : $$\mathfrak{gen}(\mathcal{A} \otimes M_{n}(\mathbb{C})) \leq \lceil 1+(r-1)/n^{2} \rceil$$
Question 3 : Is it true for von Neumann algebras ?
(N. Ozawa: "yes") $\to$ (reference or proof ?)
Theorem (here p 3): Free interpolated $L(\mathbb{F}_{r}) \otimes M_{n}(\mathbb{C}) \simeq L(\mathbb{F}_{1+(r-1)/n^{2}})$
Remark : See the correspondence with "$1+(r-1)/n^{2}$" for the two previous theorems.
Corollary : $L(\mathbb{F}_{n}) \simeq L(\mathbb{F}_{(n-1)k^{2}+1}) \otimes M_{k}(\mathbb{C})$ (in particular $L(\mathbb{F}_{2}) \simeq L(\mathbb{F}_{5}) \otimes M_{2}(\mathbb{C})$).
Definition: A $C^{*}$-algebra $\mathcal{A}$ is "connected" if it does not contain nontrivial projections.
Note that the commutative $C^{*}$-algebra $ C(X)$ is "connected" iff $X$ is connected.
Kadison-Kaplansky conjecture : for every torsion-free discrete group $\Gamma$, $C^{*}_{r}(\Gamma)$ is "connected".
Remark: $C^{*}_{r}(\mathbb{F}_{n}) \not\simeq C^{*}_{r}(\mathbb{F}_{(n-1)k^{2}+1}) \otimes M_{k}(\mathbb{C})$ because $C^{*}_{r}(\mathbb{F}_{n}) $ is "connected".
Nevertheless, they are von Neumann equivalent.
Question 4 :
- Is there a (unique) "connected" $C^{*}$-algebra, von Neumann equivalent to the $C^{*}$-algebra $pM_{n}(C^{*}_{r}(\mathbb{F}_{m}) )p$, with $p \in M_{n}(L(\mathbb{F}_{m}) )$ a projection ?
- Is its isomorphism class given by $s = (m-1)/r^{2}+1$ with $r=n \tau(p)$?
- If so, we call it the free interpolated $C^{*}$-algebra $C^{*}_{r}(\mathbb{F}_{s}) $. What's its $K$-theory ?
(This question has been improved after N. Ozawa's comment)
Remark : $\mathfrak{gen}(C^{*}_{r}(\mathbb{F}_{n})) \leq 2n$ then $\mathfrak{gen}(C^{*}_{r}(\mathbb{F}_{n}) \otimes M_{m}(\mathbb{C})) \leq 2$, for $m \geq \sqrt{2n-1}$, and so $C^{*}_{r}(\mathbb{F}_{n}) \otimes M_{n}(\mathbb{C})$ is singly generated.
Consequence : $L(\mathbb{F}_{n}) \otimes M_{n}(\mathbb{C})$ is also singly generated.
If the free group isomorphism problem is true, then $L(\mathbb{F}_{n}) \simeq L(\mathbb{F}_{2})$ is singly generated.
Summary : "$L(\mathbb{F}_{n}) \simeq L(\mathbb{F}_{m})$" $ \Rightarrow $ "$L(\mathbb{F}_{n})$ is singly generated", $\forall n, m \geq 2$
Is the converse known to be true ?
Question 5 : "$L(\mathbb{F}_{n})$ is singly generated" $ \Rightarrow $ "$L(\mathbb{F}_{n}) \simeq L(\mathbb{F}_{m})$" , $\forall n, m \geq 2$ ?
(This question has been improved after N. Ozawa's comment)
Remark (thanks to Jon Bannon comment below):
The paper Generator of $II_{1}$ factor of Dykema-Sinclair-Smith-White, focuses on the properties of the invariant $\mathcal{G}(N)$ introduced by Junhao Shen. In particular: if $\mathcal{G}(L(\mathbb{F}_{r}))>0$ for any particular $r>1$, then the free group factors are not isomorphic and are not singly generated for $r$ large enough.
So if $L(\mathbb{F}_{r})$ is singly generated $\forall r>1$, then $\mathcal{G}(L(\mathbb{F}_{r})) = 0$. Does this imply $L(\mathbb{F}_{n}) \simeq L(\mathbb{F}_{m})$ ?
