A stabilization theorem for Banach algebras
The paper deals with Bass' stable rank of a (real or complex) Banach algebra. The result mentioned in the title is the following: the sequence \(\{\pi_ i(GL_ m(A))\}_ m\) stabilizes when \(m\geq sr(A)+i+1\) where sr(A) is Bass' stable rank of A. This means that \(K_ i^{top}(A)=\pi_{i-1}(GL_ m(A))\) for those m. The proof is based on a theorem which has some interest in itself: if X is a compact space with topological dimension d and A(X) denotes the Banach algebra of all maps \(X\to A\), then \(sr(A(X))\leq sr(A)+d.\) For related results and methods, see \textit{M. Rieffel} [Proc. Lond. Math. Soc., III. Ser. 46, 301-333 (1983; Zbl 0533.46046)] and \textit{H. Schröder} [Math. Ann. 267, 271-277 (1984; Zbl 0525.46040)].
scientific article; zbMATH DE number 3973122