Geometry of unitaries in a finite algebra: variation formulas and convexity

Given a C∗-algebra A with trace τ, we compute the first and second variation formulas for the p-energy functional Fp of the unitary group UA of A, for p = 2n an even integer, namely: Fp(γ) = Z b a τ([ ˙γ∗γ˙ ] n)dt, where γ(t) ∈ UA is a smooth curve for t ∈ [a, b]. As an application of these formulas...

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Authors: Andruchow, Esteban, Recht, Lázaro
Format: article
Status:Published version
Publication Date:2008
Country:Argentina
Institution:Consejo Nacional de Investigaciones Científicas y Técnicas
Repository:CONICET Digital (CONICET)
Language:English
OAI Identifier:oai:ri.conicet.gov.ar:11336/19467
Online Access:http://hdl.handle.net/11336/19467
Access Level:Open access
Keyword:Unitary Operator
Convexity
https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
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spelling Geometry of unitaries in a finite algebra: variation formulas and convexityAndruchow, EstebanRecht, LázaroUnitary OperatorConvexityhttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1Given a C∗-algebra A with trace τ, we compute the first and second variation formulas for the p-energy functional Fp of the unitary group UA of A, for p = 2n an even integer, namely: Fp(γ) = Z b a τ([ ˙γ∗γ˙ ] n)dt, where γ(t) ∈ UA is a smooth curve for t ∈ [a, b]. As an application of these formulas, we prove that if dp denotes the geodesic distance of the Finsler metric induced by the p-norm xp = τ([x∗x] n)1/p, u0, u1, u2 ∈ UA with ui − uj < 1 2 p2 − √2 and δ(t) is a geodesic of UA joining δ(0) = u0 and δ(1) = u1, then the mapping f(t) = dp(u2, δ(t))p, t ∈ [0, 1] 15 is convex.Fil: Andruchow, Esteban. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderon; ArgentinaFil: Recht, Lázaro. Universidad Simón Bolivar; Venezuela. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderon; ArgentinaWorld Scientific2008-12info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/19467Andruchow, Esteban; Recht, Lázaro; Geometry of unitaries in a finite algebra: variation formulas and convexity; World Scientific; International Journal Of Mathematics; 19; 10; 12-2008; 1-24; 12230129-167XCONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/http://www.worldscientific.com/doi/abs/10.1142/S0129167X08005102?journalCode=ijminfo:eu-repo/semantics/altIdentifier/doi/10.1142/S0129167X08005102info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2024-05-08T13:48:19Zoai:ri.conicet.gov.ar:11336/19467instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982024-05-08 13:48:19.538CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Geometry of unitaries in a finite algebra: variation formulas and convexity
title Geometry of unitaries in a finite algebra: variation formulas and convexity
spellingShingle Geometry of unitaries in a finite algebra: variation formulas and convexity
Andruchow, Esteban
Unitary Operator
Convexity
https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
title_short Geometry of unitaries in a finite algebra: variation formulas and convexity
title_full Geometry of unitaries in a finite algebra: variation formulas and convexity
title_fullStr Geometry of unitaries in a finite algebra: variation formulas and convexity
title_full_unstemmed Geometry of unitaries in a finite algebra: variation formulas and convexity
title_sort Geometry of unitaries in a finite algebra: variation formulas and convexity
dc.creator.none.fl_str_mv Andruchow, Esteban
Recht, Lázaro
author Andruchow, Esteban
author_facet Andruchow, Esteban
Recht, Lázaro
author_role author
author2 Recht, Lázaro
author2_role author
dc.subject.none.fl_str_mv Unitary Operator
Convexity
https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
topic Unitary Operator
Convexity
https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
description Given a C∗-algebra A with trace τ, we compute the first and second variation formulas for the p-energy functional Fp of the unitary group UA of A, for p = 2n an even integer, namely: Fp(γ) = Z b a τ([ ˙γ∗γ˙ ] n)dt, where γ(t) ∈ UA is a smooth curve for t ∈ [a, b]. As an application of these formulas, we prove that if dp denotes the geodesic distance of the Finsler metric induced by the p-norm xp = τ([x∗x] n)1/p, u0, u1, u2 ∈ UA with ui − uj < 1 2 p2 − √2 and δ(t) is a geodesic of UA joining δ(0) = u0 and δ(1) = u1, then the mapping f(t) = dp(u2, δ(t))p, t ∈ [0, 1] 15 is convex.
publishDate 2008
dc.date.none.fl_str_mv 2008-12
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11336/19467
Andruchow, Esteban; Recht, Lázaro; Geometry of unitaries in a finite algebra: variation formulas and convexity; World Scientific; International Journal Of Mathematics; 19; 10; 12-2008; 1-24; 1223
0129-167X
CONICET Digital
CONICET
url http://hdl.handle.net/11336/19467
identifier_str_mv Andruchow, Esteban; Recht, Lázaro; Geometry of unitaries in a finite algebra: variation formulas and convexity; World Scientific; International Journal Of Mathematics; 19; 10; 12-2008; 1-24; 1223
0129-167X
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/url/http://www.worldscientific.com/doi/abs/10.1142/S0129167X08005102?journalCode=ijm
info:eu-repo/semantics/altIdentifier/doi/10.1142/S0129167X08005102
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv World Scientific
publisher.none.fl_str_mv World Scientific
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
repository.name.fl_str_mv CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas
repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
_version_ 1799195293703471104
score 15,81155

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