Generation of the special linear group by elementary matrices in some measure Banach algebras

Abstract

For a commutative unital ring $R$, and $n\in \mathbb{N}$, let ${\text{SL}}_n(R)$ denote the special linear group over $R$, and ${\text{E}}_n(R)$ 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 shown that ${\text{SL}}_n(A)={\text{E}}_n(A)$ for Banach subalgebras $A$ of ${\mathcal{M}}^{+}$ that are closed under the operation ${\mathcal{M}}^{+}\ni \mu \mapsto {\mu }_t$, $t\in [0,1]$, 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. Many illustrative examples of such Banach algebras $A$ are given. An example of a Banach subalgebra $A\subset {\mathcal{M}}^{+}$, that does not possess the closure property above, but for which ${\text{SL}}_n(A)={\text{E}}_n(A)$ neverthess holds, is also given.

21 pages, 1 figure. Some typos corrected