Factorization in $SL_n(R)$ with elementary matrices when $R$ is the disk algebra and the Wiener algebra

TL;DR

This paper demonstrates that the special linear group over the disk and Wiener algebras is generated by elementary matrices, leveraging a significant prior result to establish a fundamental algebraic property.

Contribution

It shows that $SL_n(R)$ is generated by elementary matrices for $R$ as the disk or Wiener algebra, extending understanding of algebraic structures over these function algebras.

Findings

$SL_n(R)$ is generated by elementary matrices.

The result relies on a deep theorem by Ivarsson and Kutzschebauch.

Applicable to $R$ as the disk algebra and Wiener algebra.

Abstract

Let $R$ be the polydisc algebra or the Wiener algebra. It is shown that the group $SL_n(R)$ is generated by the subgroup of elementary matrices with all diagonal entries $1$ and at most one nonzero off-diagonal entry. The result an easy consequence of the deep result due to Ivarsson and Kutzschebauch (Ann. of Math. 2012).