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 $\textrm{SL}_n(R)$ denote the special linear group over $R$, and $\textrm{E}_n(R)$ 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 $\textrm{SL}_n(A)=\textrm{E}_n(A)$ for Banach subalgebras $A$ of ${\mathcal{M}}^+$ that are closed under the operation ${\mathcal{M}}^+\owns μ\mapsto μ_t$, $t\in [0,1]$, where $μ_t(E):=\int_E (1-t)^x dμ(x)$ for $t\in [0,1)$, and Borel subsets $E$ of $[0,+\infty)$, and $μ_1:=μ(\{0\})δ$, where $δ\in {\mathcal{M}}^+$ 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 $\textrm{SL}_n(A)=\textrm{E}_n(A)$ neverthess holds, is also given.