Similarity implies homotopy of idempotents in Banach algebras of stable rank one
According to \textit{M. A. Rieffel} [``Dimension and stable rank in the \(K\)-theory of \(C^{*}\)-algebras'', Proc. Lond. Math. Soc. (3) 46, 301--333 (1983; Zbl 0533.46046)] a real Banach algebra \(A\) has stable rank one if and only if the set \(A^{-1}\) of invertible elements in \(A\) on dense in \(A\). In this paper the latter property is considered as a definition and the following results is proved: In a real Banach algebra of stable rank one, any two similar idempotents can be connected by a piecewise affine homotopy consisting of at most 3 affine steps.
scientific article; zbMATH DE number 5135802