Regular boundary elements

Abstract

In Banach algebra the regular elements in the closure the group invertibles are in generalized sense of index zero. 1. If A is ring, with identity 1 and invertible group A-', or more generally [6] an additive category, we shall write (1.0.1) 7= { aEA: aEaAa} for the regular or relatively Fredholm elements A, and (1.0.2) A-1A = AA-1 = { E A: E aA-la} for the decomposably regular or relatively Weyl elements A; these include the invertible elements A-', and also the idempotents (1.0.3) A={aEA:a2=a}. 1.1 THEOREM. If A is Banach algebra then (l1.1 .1) A-1A =A n cl(A-'). PROOF. The decomposably regular elements are always regular; if A is normed algebra and if E A is decomposably regular then we can write (1.1.2) = cp = qc withc EA-', p EA, q EA; we have anticipated this in the notation for (1.0.2). Putting for each n (1.1.3) bn = c(p + (1/n)(1 -p)), bn = (p + n (1-p)) c-' gives (1.1.4) Ila-bn 1J-O and bnbn = 1 = bnbn, so that also E cl(A-'). Conversely, without restriction on A, we claim that it is sufficient, for E A to be decomposably regular, that there are a' E A and b E A for which (1.1.5) = aa'a and a' = a'aa' and b E A-1 and 1 +(b-a)a' e A-1. Indeed if (1.1.5) holds and we define (1.1.6) a = a' +(1 a'a)b (1 aa') Received by the editors December 4, 1985. 1980 Mathematics Subject Classification. Primary 47B30, 46B30; Secondary 47A10, 47A53.