Projective geometry of the Poincaré disk of a C*-algebra

Fil: Andruchow, Esteban. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina.

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Autores principales: Andruchow, Esteban, Corach, Gustavo, Recht, Lázaro
Formato: Artículo publishedVersion
Lenguaje:Inglés
Publicado: Theta Foundation 2024
Materias:
Projective line
Poincaré disk
C*-algebra
Acceso en línea:http://repositorio.ungs.edu.ar:8080/xmlui/handle/UNGS/1811
Aporte de:
Repositorio Institucional Digital de Acceso Abierto (UNGS) de Universidad Nacional de General Sarmiento
id I71-R177-UNGS-1811
record_format dspace
spelling I71-R177-UNGS-18112024-12-23T13:21:48Z Projective geometry of the Poincaré disk of a C*-algebra Andruchow, Esteban Corach, Gustavo Recht, Lázaro Projective line Poincaré disk C*-algebra 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. Fil: Corach, Gustavo. Consejo Nacional de Investigaciones Científicas y Técnicas. Instituto Argentino de Matemática "Alberto P. Calderón"; Argentina. Fil: Recht, Lázaro. Consejo Nacional de Investigaciones Científicas y Técnicas. Instituto Argentino de Matemática "Alberto P. Calderón"; Argentina. We study the Poincar´e disk $d={aina: |a|<1}$ of a C$^*$-algebra $a$ from a projective point of view: $d$ is regarded as an open subset of the projective line $pa$, the space of complemented rank one submodules of $a^2$. We introduce the concept of cross ratio of four points in $pa$. Our main result establishes the relation between the exponential map $Exp_{z_0}(z_1)$ of $d$ ($z_0,z_1ind$) and the cross ratio of the four-tuple $$delta(-infty), delta(0)=z_0, delta(1)=z_1 , delta(+infty),$$ where $delta$ is the unique geodesic of $d$ joining $z_0$ and $z_1$ at times $t=0$ and $t=1$, respectively. Here $delta(-infty)=lim_{to-infty}delta(t)$ and $delta(+infty)=lim_{to+infty}delta(t)$, the limits are considered in the strong operator topology, and may take values in the universal algebra $a^{**}$. 2024-12-23T13:21:48Z 2024-12-23T13:21:48Z 2023 info:eu-repo/semantics/article info:ar-repo/semantics/artículo info:eu-repo/semantics/publishedVersion Andruchow, E., Corach, G. y Recht, L. (2023). Projective geometry of the Poincaré disk of a C*-algebra. Journal of Operator Theory, 89(1), 155-182. 0379-4024 http://repositorio.ungs.edu.ar:8080/xmlui/handle/UNGS/1811 eng http://dx.doi.org/10.7900/jot.2021dec27.2356 info:eu-repo/semantics/restrictedAccess https://creativecommons.org/licenses/by-nc-nd/4.0/ application/pdf Theta Foundation Journal of Operator Theory. 2023; 89(1): 155-182 http://www.mathjournals.org/jot/2023-089-001/
institution Universidad Nacional de General Sarmiento
institution_str I-71
repository_str R-177
collection Repositorio Institucional Digital de Acceso Abierto (UNGS)
language Inglés
orig_language_str_mv eng
topic Projective line
Poincaré disk
C*-algebra
spellingShingle Projective line
Poincaré disk
C*-algebra
Andruchow, Esteban
Corach, Gustavo
Recht, Lázaro
Projective geometry of the Poincaré disk of a C*-algebra
topic_facet Projective line
Poincaré disk
C*-algebra
description Fil: Andruchow, Esteban. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina.
format Artículo
Artículo
publishedVersion
author Andruchow, Esteban
Corach, Gustavo
Recht, Lázaro
author_facet Andruchow, Esteban
Corach, Gustavo
Recht, Lázaro
author_sort Andruchow, Esteban
title Projective geometry of the Poincaré disk of a C*-algebra
title_short Projective geometry of the Poincaré disk of a C*-algebra
title_full Projective geometry of the Poincaré disk of a C*-algebra
title_fullStr Projective geometry of the Poincaré disk of a C*-algebra
title_full_unstemmed Projective geometry of the Poincaré disk of a C*-algebra
title_sort projective geometry of the poincaré disk of a c*-algebra
publisher Theta Foundation
publishDate 2024
url http://repositorio.ungs.edu.ar:8080/xmlui/handle/UNGS/1811
work_keys_str_mv AT andruchowesteban projectivegeometryofthepoincarediskofacalgebra
AT corachgustavo projectivegeometryofthepoincarediskofacalgebra
AT rechtlazaro projectivegeometryofthepoincarediskofacalgebra
_version_ 1824528660904804352

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