An explicit construction of the universal division ring of fractions of

  • Andrei Jaikin-Zapirain

    Universidad Autónoma de Madrid, Spain
An explicit construction of the universal division ring of fractions of $E\langle\langle x_1,\ldots, x_d\rangle \rangle$ cover

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Abstract

We give a sufficient and necessary condition for a regular Sylvester matrix rank function on a ring to be equal to its inner rank . 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 are operators in a finite von Neumann algebra with a faithful normal trace , then they generate the free division ring on in the algebra of unbounded operators affiliated to if and only if the space of tuples of finite rank operators on 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 , where is a division ring, and we use it in order to show the following instance of pro- Lück approximation.

Let be a finitely generated free pro -group, a chain of normal open subgroups of with trivial intersection and a matrix over . Denote by the matrix over obtained from the matrix by applying the natural homomorphism . 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 of the matrix .

Cite this article

Andrei Jaikin-Zapirain, An explicit construction of the universal division ring of fractions of . J. Comb. Algebra 4 (2020), no. 4, pp. 369–395

DOI 10.4171/JCA/47