On the geometry of generalized inverses

Abstract

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 isometrics, 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 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.