In this post I would like to ask several of questions related to Dixmier problem. I will try to make the post as self-contained as possible.
A discrete group $G$ is unitarisable if for every Hilbert space $H$ and a homomorphism $\pi:G\rightarrow B(H)$ such that $||\pi(g)||<C$ for every $g\in G$ and some constant $C$ there exists an invertible operator $S\in B(H)$ such that for every $g\in G$ we have that $S\pi(g)S^{-1}$ is unitary operator. Note that the unitarisability property passes to subgroups.
The following is still open:
Dixmier problem: $G$ is amenable iff $G$ is unitarisable.
Denote by $T_1(G)$ the space of functions $f:G\rightarrow \mathbb{C}$ with the following norm:
$$||f||_{T_1(G)}=\inf \{ \sup\limits_{s}\sum\limits_{t}f_1(s,t)+\sup\limits_{t}\sum\limits_{s}f_2(s,t)\}$$
where $\inf$ is taken over all decompositions of $f(s^{-1}t)=f_1(s,t)+f_2(s,t)$.
It was proved by Bozejko and Fendler that is $G$ is unitarisable then $T_1(G)\subseteq l_2(G)$, equivalently, there exist a constant $C>0$ such that for every $f\in \mathbb{C}[G]$ we have $$||f||_{l_2(G)}\leq C ||f||_{T_1(G)}.$$
From this result it is immediate that $\mathbb{F}_{\infty}$ is not unitarisable. Indeed, let $f:\mathbb{F}_{\infty}\rightarrow\mathbb{C}$ be the characteristic function of the words of length $1$ with respect to the standard set of generators. Then $||f||_{l_2(G)}=\infty$. Since $f(s^{-1}t)=1_{\{(s,t): |s|>|t|, |st|=1\}}+1_{\{(s,t): |s|<|t|, |st|=1\}}$ we have $||f||_{T_1(G)}\leq 2$.
Question 1: Let $G$ be such that $T_1(G)\not\subset l_{2}(G)$. Is it true that there exists a characteristic function of an infinite set $S\subset G$ such that its $T_1(G)$ norm of it is bounded? Are there sets $S_n$ such that $|S_n|\rightarrow \infty$ and $||1_{S_n}||_{T_1(G)}\leq C$ for some constant $C>0$?
Note that Bozejko-Fendler result helps to catch examples of non-unitarisable group without free subgroups. The last fact is combination of recent results of Epstein, Monod and Osin.
It is not clear however if in their example the function that violates Bozejko-Fendler condition can be chosen to be characteristic.
Is the following true:
Question 2: $G$ is not amenable iff there exists an infinite set $R\subset G$ such that $\Delta(R)=\{(s,t)\in G\times G: s^{-1}t\in R\}$ and $\Delta(R)=R_1\cup R_2$ with $R_1\cap R_2=\emptyset$ and $|\{s:(s,t)\in R_1\}|+|\{t:(s,t)\in R_2\}|<C$ for some constant $C>0$ and all $s,t\in G$.
or, maybe, we just have a positive answer to the following question:
Question 3: Assume $G$ satisfies the second part of the Question 2. Namely, there exists an infinite set $R\subset G$ such that $\Delta(R)=\{(s,t)\in G\times G: s^{-1}t\in R\}$ and $\Delta(R)=R_1\cup R_2$ with $R_1\cap R_2=\emptyset$ and $|\{s:(s,t)\in R_1\}|+|\{t:(s,t)\in R_2\}|<C$ for some constant $C>0$ and all $s,t\in G$.
Is it true that $G$ contains the free group on $2$ generators?
Question 2 can be restated as follows:
Question 2': $G$ is amenable iff there exist a sequence of subsets $S_n\subset G$ with $|S_n|\rightarrow \infty$ a constant $C>0$ such that for every finite sets $A,B\subset G$ we have $$|\Delta(S_n)\cap A\times B|\leq C (|A|+|B|)$$
Note that, in the Q. 2' one can take $A=B$.
the same for the Question 3:
Question 3': Is it true that if $G$ satisfies the second condition in the Question 2' then $\mathbb{F}_2$ is a subgroup of $G$?
Disclaimer: Some of the questions above were communicated to me by Gilles Pisier in discussions following my talk on his working group seminar.
