6 problems

Let $\mathbb{K}$ be a subfield of $\mathbb{C}$, and let $G$ be a countable group with a bound on the orders of its finite subgroups. For a finite matrix $A$ over $\mathbb{K}[G]$, l…

Let $\mathbb{K}$ be a subfield of $\mathbb{C}$, let $G$ be a countable group with a bound on the orders of its finite subgroups, and let $\mathrm{lcm}(G)$ denote the least common m…

Let $G$ be a group in the class $\mathcal{T}$ with finite $\operatorname{lcm}(G)$, and let $K\subseteq\mathbb{C}$ be any subfield closed under complex conjugation. Let…

Let $G$ be a torsion-free countable group, and let ${\mathcal{D}}(\mathbb{C}G\subset {\mathcal{U}}G)$ denote the ring appearing in the Atiyah-conjecture construction. Atiyah conjec…

Let $G$ be a torsion-free group, let $m,n$ be positive integers, and let $A\in M_{m,n}(\mathbb{Q}G)$. Write $r_A\colon \mathcal{N}(G)^m\to\mathcal{N}(G)^n$ for right multiplication…

Analytic zero-divisor conjecture. If $0 \neq \alpha \in \mathbb{C}G$ and $0 \neq f \in \ell^2(G)$, then