On the geometry of generalized inverses

We 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, G A × G A and U A × U A , where G A is the group of invertibles of A and U A 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 ℐ p of p in ℐ is simply connected. If 1 – p is also purely infinite, then ℐ p is contractible. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)