Essentially commuting projections

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 com...

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Autores principales: Andruchow, Esteban, Chiumiento, Eduardo Hernán, Di Iorio y Lucero, M. E.
Formato: Articulo
Lenguaje:Inglés
Publicado: 2015
Materias:
Matemática
Compact operators
Fredholm index
Geodesics
Projections
Acceso en línea:http://sedici.unlp.edu.ar/handle/10915/87080
Aporte de:
SEDICI (UNLP) de Universidad Nacional de La Plata
id I19-R120-10915-87080
record_format dspace
institution Universidad Nacional de La Plata
institution_str I-19
repository_str R-120
collection SEDICI (UNLP)
language Inglés
topic Matemática
Compact operators
Fredholm index
Geodesics
Projections
spellingShingle Matemática
Compact operators
Fredholm index
Geodesics
Projections
Andruchow, Esteban
Chiumiento, Eduardo Hernán
Di Iorio y Lucero, M. E.
Essentially commuting projections
topic_facet Matemática
Compact operators
Fredholm index
Geodesics
Projections
description 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.
format Articulo
Articulo
author Andruchow, Esteban
Chiumiento, Eduardo Hernán
Di Iorio y Lucero, M. E.
author_facet Andruchow, Esteban
Chiumiento, Eduardo Hernán
Di Iorio y Lucero, M. E.
author_sort Andruchow, Esteban
title Essentially commuting projections
title_short Essentially commuting projections
title_full Essentially commuting projections
title_fullStr Essentially commuting projections
title_full_unstemmed Essentially commuting projections
title_sort essentially commuting projections
publishDate 2015
url http://sedici.unlp.edu.ar/handle/10915/87080
work_keys_str_mv AT andruchowesteban essentiallycommutingprojections
AT chiumientoeduardohernan essentiallycommutingprojections
AT diiorioylucerome essentiallycommutingprojections
bdutipo_str Repositorios
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