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

Sasane, Amol (2023) Generation of the special linear group by elementary matrices in some measure Banach algebras. Studia Mathematica, 270 (1). pp. 1-16. ISSN 0039-3223

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Abstract

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.

Item Type:	Article
Official URL:	https://www.impan.pl/en/publishing-house/journals-...
Additional Information:	© 2022 Instytut Matematyczny PAN
Divisions:	Mathematics
Subjects:	Q Science > QA Mathematics
Date Deposited:	18 Feb 2022 10:06
Last Modified:	15 Apr 2024 18:36
URI:	http://eprints.lse.ac.uk/id/eprint/113779

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