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 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. Gramsch [16] studied the set of Fredholm operators with fixed dimension of the kernel, and proved that it is an analytic homogeneous manifold. He also extended some of these results to the context of Frechet algebras with open group of invertible elements. In [12] there is a geometrical study of certain parts of S. Observe that (a, b) belongs to S if and only if aba = a and bab = b; in particular ab and ba belong to Q, the set of idempotents of A. For a fixed r in Q, consider the set S r = {(a, b) ∈ S : ar = a, rb = b, ba = r}. There is a natural action of G A over S r given by *

Geometry of $\mathfrak {I}$-Stiefel manifolds

“…There are several papers concerning the geometry and topology of partial isometries endowed with the spectral norm (see, for instance, [2], [3], [9] and [12]) . On the other hand, the article [1] is devoted to studying partial isometries of finite rank with the Hilbert-Schmidt norm.…”

Local Minimal Curves in Homogeneous Reductive Spaces of the Unitary Group of a Finite von Neumann Algebra

“…On the other hand, since we have γ(t 2 ) ∈ V R (ρ), by Remark 3.4 there exists a unique z 2 ∈ B R (0) such that L e z 2 ρ = γ(t 2 ). Using our assumption that R ≤ r, we have z 2 = z 2 , and then δ 2 is the unique geodesic in V R (ρ) joining ρ and γ(t 2 ). An easy computation shows that L q , ∞ (δ 2 ) = z 2 .…”

“…if two partial isometries lie closer than 1/2 in the operator norm, then they are conjugate by a pair of unitaries. In [2] was proved that each connected component, which as a consequence of local transitivity coincides with an orbit, is a homogeneous space of U M ×U M and a C ∞ submanifold of M. Hence, by Remark 2.1, to give a reductive structure we just have to prove the orthogonality condition.…”

Metric geometry of partial isometries in a finite von Neumann algebra

“…The action of U M on I p is locally transitive: if v, v 0 ∈ I p with v − v 0 < 1/2, then there exists a unitary valued C ∞ map ω = ω(v, v 0 ) such that ωv 0 = v (see [14,3] and [2] for an account of these facts). In the case of a general von Neumann algebra the action is not transitive.…”