A note on geodesics of projections in the Calkin algebra

• Autores: Andruchow, Esteban
• Año de publicación: 2020
• Idioma: inglés
• Tipo de recurso: artículo
• Fuente: Archiv Der Mathematik. Nov. 2020; 115(5): 545-553; https://link.springer.com/journal/13/volumes-and-issues/115-5
• Materia: Calkin algebra; Geodesics of projections; Projections
• Institución: Universidad Nacional de General Sarmiento

Fil: Andruchow, Esteban. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina.
Fil: Andruchow, Esteban. Consejo Nacional de Investigaciones Científicas y Técnicas. Instituto Argentino de Matemática "Alberto P. Calderón"; Argentina.
Let C(H) = B(H) / K(H) be the Calkin algebra (B(H) the algebra of bounded operators on the Hilbert space H, K(H) the ideal of compact operators, and π: B(H) → C(H) the quotient map), and PC(H) the differentiable manifold of selfadjoint projections in C(H). A projection p in C(H) can be lifted to a projection P∈ B(H) : π(P) = p. We show that, given p, q∈ PC(H), there exists a minimal geodesic of PC(H) which joins p and q if and only if there exist lifting projections P and Q such that either both N(P- Q± 1) are finite dimensional, or both are infinite dimensional. The minimal geodesic is unique if p+ q- 1 has trivial anhihilator. Here the assertion that a geodesic is minimal means that it is shorter than any other piecewise smooth curve γ(t) ∈ PC(H), t∈ I, joining the same endpoints, where the length of γ is measured by ∫ I‖ γ˙ (t) ‖ dt.