Generation of the special linear group by elementary matrices in some measure Banach algebras
Volume 270 / 2023
Abstract
For a commutative unital ring $R$, and $n\in \mathbb N$, let $\mathrm{SL}_n(R)$ denote the special linear group over $R$, and $\mathrm E_n(R)$ the subgroup of elementary matrices. Let $\mathcal M^+$ be the Banach algebra of all complex Borel measures on $[0,+\infty )$ with the norm given by the total variation, the usual operations of addition and scalar multiplication, and with convolution. It is first shown that $\mathrm{SL}_n(A)=\mathrm E_n(A)$ for Banach subalgebras $A$ of $\mathcal M^+$ that are closed under the operation $\mathcal M^+\owns \mu \mapsto \mu _t$, $t\in [0,1]$, where $\mu _t(E):=\int _E (1-t)^x \,d\mu (x)$ for $t\in [0,1)$, and Borel subsets $E$ of $[0,+\infty )$, and $\mu _1:=\mu (\{0\})\delta $, where $\delta \in \mathcal M^+$ is the Dirac measure. Using this, and with auxiliary results established in the article, many illustrative examples of such Banach algebras $A$ are given, including several well-studied classical Banach algebras such as the class of analytic almost periodic functions. An example of a Banach subalgebra $A\subset \mathcal M^+$ that does not have the closure property above, but for which $\mathrm{SL}_n(A)=\mathrm E_n(A)$ nevertheless holds, is also constructed.
