AbstractWe study the set S = {(a, b) ∈ A × A : aba = a, bab = b} which pairs the relatively regular elements of a Banach algebra A with their pseudoinverses, and prove that it is an analytic submanifold of A × A. If A is a C*‐algebra, inside S lies a copy the set ℐ of partial isometries, we prove that this set is a C∞ submanifold of S (as well as a submanifold of A). These manifolds carry actions from, respectively, GA × GA and UA × UA, where GA is the group of invertibles of A and UA is the subgroup of unitary elements. These actions define homogeneous reductive structures for S and ℐ (in the differential geometric sense). Certain topological and homotopical properties of these sets are derived. In particular, it is shown that if A is a von Neumann algebra and p is a purely infinite projection of A, then the connected component ℐp of p in ℐ is simply connected. If 1 – p is also purely infinite, then ℐp is contractible. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)