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 aBanach 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.