An explicit construction of the universal division ring of fractions of $E⟨⟨x_1,\dots ,x_d⟩⟩$

Abstract

We give a sufficient and necessary condition for a regular Sylvester matrix rank function on a ring $R$ to be equal to its inner rank ${\rho }_R$. We apply it in two different contexts.

In our first application, we reprove a recent result of T. Mai, R. Speicher and S. Yin: if $X_1,\dots ,X_d$ are operators in a finite von Neumann algebra $\mathcal{M}$ with a faithful normal trace $\tau$, then they generate the free division ring on $X_1,\dots ,X_d$ in the algebra of unbounded operators affiliated to $\mathcal{M}$ if and only if the space of tuples $(T_1,\dots ,T_d)$ of finite rank operators on $L^2(\mathcal{M},\tau )$ satisfying [ \sum_{i=1}^d [T_k,X_k]=0, ] is trivial.

In our second and main application we construct explicitly the universal division ring of fractions of $E⟨⟨x_1,\dots ,x_n⟩⟩$, where $E$ is a division ring, and we use it in order to show the following instance of pro-$p$ Lück approximation.

Let $F$ be a finitely generated free pro $p$-group, $F=F_1>F_2>\cdots$ a chain of normal open subgroups of $F$ with trivial intersection and $A$ a matrix over ${\mathbb{F}}_p[[F]]$. Denote by $A_i$ the matrix over ${\mathbb{F}}_p[F/F_i]$ obtained from the matrix $A$ by applying the natural homomorphism ${\mathbb{F}}_p[[F]]\to {\mathbb{F}}_p[F/F_i]$. Then there exists the limit [ \displaystyle \lim_{i\to \infty} \frac{\mathrm {rk}_{\mathbb F_p} (A_i)}{|F

|} ] and it is equal to the inner rank ${\rho }_{{\mathbb{F}}_p[[F]]}(A)$ of the matrix $A$.