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

Abstract

For a commutative unital ring R, and n P N, let SL n pRq denote the special linear group over R, and E n pRq the subgroup of elementary matrices. Let M `be the Banach algebra of all complex Borel measures on r0, `8q with the norm given by the total variation, the usual operations of addition and scalar multiplication, and with convolution. It is shown that SL n pAq " E n pAq for Banach subalgebras A of M `that are closed under the operation M `Q µ Þ Ñ µ t , t P r0, 1s, where µ t pEq :" ş E p1 ´tq x dµpxq for t P r0, 1q, and Borel subsets E of r0, `8q, and µ 1 :" µpt0uqδ, where δ P M `is the Dirac measure. Many illustrative examples of such Banach algebras A are given. An example of a Banach subalgebra A Ă M `, that does not possess the closure property above, but for which SL n pAq " E n pAq nevertheless holds, is also given.