Stable range in Banach algebras (Q1061983)

Let Sr A denote the stable range of a Banach algebra A. The author relates the stable range conditions on A with the topological structure of A and gives a sufficient condition to have Sr \(A\leq n\). As a consequence of these results he obtains the stable range of some Banach algebras, deduces the known result that Sr B(H)\(=\infty\) if H is an infinite dimensional Hilbert space an gives an extension of another known result that the invertible operators on H are not dense in B(H). In the last section a characterization theorem is given for the validity of Sr \(A\leq n\) when A is commutative and it is shown that Sr A\(=[d]+1\) when A is regular with d the covering dimension of its maximal ideal space.