Essentially commuting projections

• Autores: Andruchow, Esteban; Chiumiento, Eduardo Hernan; Di Iorio y Lucero, María Eugenia
• Año de publicación: 2015
• Idioma: inglés
• Tipo de recurso: artículo
• Materia: Projections; Compact Operators; Fredholm Index; Geodesics
• Institución: Consejo Nacional de Investigaciones Científicas y Técnicas

Let H=H+⊕H- be a fixed orthogonal decomposition of a Hilbert space, with both subspaces of infinite dimension, and let E+, E- be the projections onto H+ and H-. We study the set Pcc of orthogonal projections P in H which essentially commute with E+ (or equivalently with E-), i.e.[P,E+]=PE+-E+Pis compact. By means of the projection π onto the Calkin algebra, one sees that these projections P∈Pcc fall into nine classes. Four discrete classes, which correspond to π(P) being 0, 1, π(E+) or π(E-), and five essential classes which we describe below. The discrete classes are, respectively, the finite rank projections, finite co-rank projections, the Sato Grassmannian of H+ and the Sato Grassmannian of H-. Thus the connected components of each of these classes are parametrized by the integers (via de rank, the co-rank or the Fredholm index, respectively). The essential classes are shown to be connected.We are interested in the geometric structure of Pcc, being the set of selfadjoint projections of the C*-algebra Bcc of operators in B(H) which essentially commute with E+. In particular, we study the problem of existence of minimal geodesics joining two given projections in the same component. We show that the Hopf-Rinow Theorem holds in the discrete classes, but not in the essential classes. Conditions for the existence and uniqueness of geodesics in these latter classes are found.
Fil: Andruchow, Esteban. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemáticas; Argentina. Universidad Nacional de General Sarmiento; Argentina
Fil: Chiumiento, Eduardo Hernan. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemáticas; Argentina. Universidad Nacional de La Plata. Facultad de Cencias Económicas; Argentina
Fil: Di Iorio y Lucero, María Eugenia. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemáticas; Argentina. Universidad Nacional de General Sarmiento; Argentina