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On the geometry of generalized inverses

✍ Scribed by E. Andruchow; G. Corach; M. Mbekhta

Publisher
John Wiley and Sons
Year
2005
Tongue
English
Weight
219 KB
Volume
278
Category
Article
ISSN
0025-584X

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✦ Synopsis

Abstract

We study the set S = {(a, b) ∈ A Γ— 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 Γ— 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~ Γ— G~A~ and U~A~ Γ— 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. (Β© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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