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

TL;DR

This paper proves that in certain Banach algebras of measures, the special linear group is generated by elementary matrices, extending known results to new algebraic structures with specific closure properties.

Contribution

It establishes the equality of SL_n(A) and E_n(A) for specific measure Banach algebras, including cases lacking the usual closure property.

Findings

SL_n(A) equals E_n(A) in these measure Banach algebras

Examples of algebras where the property holds without the closure condition

Provides illustrative examples and counterexamples

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 ${\mathcal{M}}^+\owns \mu \mapsto \mu_t$, $t\in [0,1]$, 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 $\textrm{SL}_n(A)=\textrm{E}_n(A)$ neverthess holds, is also given.