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 2) : 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 equivalentto 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 $L(\mathbb{F}_{n}) \simeq L(\mathbb{F}_{m})$ ?

explainyour reasons for downvoting a question. Perhaps you might like to leave suggestions for how the question could be improved. $\endgroup$ – MTS Aug 1 '13 at 16:00nowwell improved... $\endgroup$ – Sebastien Palcoux Aug 1 '13 at 16:50