Manifolds of semi-negative curvature

Abstract

The notion of nonpositive curvature in Alexandrov's sense is extended to include p-uniformly convex Banach spaces. Infinite dimensional manifolds of semi-negative curvature with a puniformly convex tangent norm fall in this class on nonpositively curved spaces, and several well-known results, such as existence and uniqueness of best approximations from convex closed sets, or the Bruhat-Tits fixed point theorem, are shown to hold in this setting, without dimension restrictions. Homogeneous spaces G/K of Banach-Lie groups of seminegative curvature are also studied, explicit estimates on the geodesic distance and sectional curvature are obtained. A characterization of convex homogeneous submanifolds is given in terms of the Banach-Lie algebras. A splitting theorem via convex expansive submanifolds is proven, inducing the corresponding splitting of the Banach-Lie group G. Finally, these notions are used to study the structure of the classical Banach-Lie groups of bounded linear operators acting on a Hilbert space, and the splittings induced by conditional expectations in such setting.

Metric geometry of infinite-dimensional Lie groups and their homogeneous spaces

“…At the end of Section 3, we discuss analogues of the second Riemannian property mentioned above, regarding geodesics of G and geodesics of M. The main results of Section 3 are Theorem 3. 26 and Theorem 3. 29.…”

“…The assertion on the speed of γ is immediate from the definitions and Theorem 3. 26. The assertion on the uniqueness can be proved using the same idea as in [26,Proposition 3.6].…”

“…Theorem 3. 26. Let G, K, M ≃ G/K be as in Definition 3.1 and µ a L-uniform Finsler metric in G. Then…”

Decompositions and complexifications of some infinite-dimensional homogeneous spaces

“…Remark 3.16. The splitting theorem of Porta and Recht [14] asserts that if we have a unital inclusion of C * -algebras B ⊆ A and a conditional expectation E : A → B then the map Φ : [5]. The CPR theorem in the context of several reductive structures (Theorem 3.14) covers for example the case of multiple unital inclusions of C * -algebras and conditional expectations between them…”

Spaces of nonpositive curvature arising from a finite algebra

“…Recently [12,14,22], we have studied spaces of perturbations of Schatten operators where again, despite the infinite dimensional setting, the metrics introduced come from the (infinite) trace of B(H). The results obtained in those works, together with a factorization theorem for the group of invertible elements of a C * -algebra obtained by Porta and Recht in [30], have been extended to Banach-Lie groups in [15]. A closer relevant precedent is [5] where the (weak) Riemannian case, corresponding to p = 2, is studied as a geodesic length space.…”