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

For a commutative unital ring R, and n ε N, let SLn(R) denote the special linear group over R, and En(R) the subgroup of elementary matrices. Let M+ be the Banach algebra of all complex Borel measures on [0,+8] with the norm given by the total variation, the usual operations of addition and scalar multiplication, and with convolution. It is first shown that SLn(A) = En(A) for Banach subalgebras A of M+ that are closed under the operation M+ ε μ → μt, t ε [0, 1], where μ1(E) := ∫E (1 - t)x dμ(x) for t ε [0, 1), and Borel subsets E of [0, +q), and μ1 := μ({0})δ, where δ ε 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 Ă M+, that does not possess the closure property above, but for which SLn(A) = En(A) nevertheless holds, is also constructed.