Banach manifolds of algebraic elements in the algebra(mathcal{L})(H) of bounded linear operatorsof bounded linear operators

✍ Scribed by José M. Isidro

• Book ID: 111487824
• Publisher: SP Versita
• Year: 2005
• Tongue: English
• Weight: 313 KB
• Volume: 3
• Category: Article
• ISSN: 1895-1074
• DOI: 10.2478/bf02479195

✦ Synopsis

Given a complex Hilbert space H, we study the manifold A of algebraic elements in Z = L(H). We represent A as a disjoint union of closed connected subsets M of Z each of which is an orbit under the action of G, the group of all C * -algebra automorphisms of Z. Those orbits M consisting of hermitian algebraic elements with a fixed finite rank r, (0 < r < ∞) are real-analytic direct submanifolds of Z. Using the C * -algebra structure of Z, a Banach-manifold structure and a G-invariant torsionfree affine connection ∇ are defined on M , and the geodesics are computed. If M is the orbit of a finite rank projection, then a G-invariant Riemann structure is defined with respect to which ∇ is the Levi-Civita connection.