3 problems

Strong 1-boundedness conjecture. The von Neumann algebra $M_{1}$ is strongly $1$-bounded.

Free entropy dimension conjecture. Then $L(\Gamma)$ is not strongly one-bounded, $\delta_{0}(\mathbb{C}\Gamma)>1$, and

Let $\Gamma$ be a group generated by $g_{1},\dots,g_{n}$ and assume that $L(\Gamma)$ can be embedded into $R^{\omega}$. Let $V\subset\ell^{2}(\Gamma)^{n}$ be the subspace … Let…