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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Formato: | Articulo |
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2015
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Acceso en línea: | http://sedici.unlp.edu.ar/handle/10915/87080 |
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I19-R120-10915-87080 |
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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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1764820489623044097 |