Push-outs of derivations

Niels Grønbæk

Push-outs of derivations

Institut for Matematiske Fag

Abstract

Lad A være en banachalgebra og lad X være et banach A-bimodul. Til beregning af H¹(A,X) er det ofte nyttigt at udvide en given derivation D:A->X til en banachalgebra B, der indeholder A som et to-sidet ideal, og herved at udnytte (eller etablerere) arvelige egenskaber. Ofte gøres dette ved hjælp af (begrænsede/ubegrænsede) approksimerende enheder, hvor udvidelsen konstrueres som en grænseværdi af operatorer b->D(ba)-b.D(a), i en passende operatortopologi. Kernen i denne konstruktion består i at vise at grænseværdioperatoren opfylder derivationsegenskaben. I dette arbejde gør vi det klart, hvad der er ananlytisk og hvad der er algabraisk i denne tilgang ved at præsentere et algebraisk skema der giver derivationer i alle situationer. For at opnå dette er man i midlertid nødt til at udvide modulet X. Vi benytter vores konstruktion til at give forbedringer og kortere beviser for en række resultater fra litteraturen og til at give en betingelse der er nødvendig og tilstrækkelig for at bifladhed og biprojektivitet arves til idealer.

Udgivelsesdato: May 2008

Originalsprog	Engelsk
Tidsskrift	Proceedings of the Indian Academy of sciences. Mathematical sciences
Vol/bind	118
Udgave nummer	2
Sider (fra-til)	235-243
ISSN	0253-4142
Status	Udgivet - 2008

Citationsformater

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abstract = "Let A be a Banach algebra and let X be a Banach A-bimodule. In studying H¹(A,X) it is often useful to extend agiven derivation D: A->X to a Banach algebra Bcontaining A as an ideal, thereby exploiting (or establishing)hereditary properties. This is usually done using (bounded/unbounded)approximate identities to obtain the extension as a limit of operatorsb->D(ba)-b.D(a), a in A, in an appropriate operator topology, themain point in the proof being to show that the limit map is in fact aderivation. In this paper we make clear which part of this approach isanalytic and which algebraic by presenting an algebraic scheme thatgives derivations in all situations at the cost of enlarging themodule. We use our construction to give improvements and shorterproofs of some resultsfrom the literatureand to give a necessary and sufficient condition that biprojectivity andbiflatness are inherited to ideals",

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N2 - Let A be a Banach algebra and let X be a Banach A-bimodule. In studying H¹(A,X) it is often useful to extend agiven derivation D: A->X to a Banach algebra Bcontaining A as an ideal, thereby exploiting (or establishing)hereditary properties. This is usually done using (bounded/unbounded)approximate identities to obtain the extension as a limit of operatorsb->D(ba)-b.D(a), a in A, in an appropriate operator topology, themain point in the proof being to show that the limit map is in fact aderivation. In this paper we make clear which part of this approach isanalytic and which algebraic by presenting an algebraic scheme thatgives derivations in all situations at the cost of enlarging themodule. We use our construction to give improvements and shorterproofs of some resultsfrom the literatureand to give a necessary and sufficient condition that biprojectivity andbiflatness are inherited to ideals

AB - Let A be a Banach algebra and let X be a Banach A-bimodule. In studying H¹(A,X) it is often useful to extend agiven derivation D: A->X to a Banach algebra Bcontaining A as an ideal, thereby exploiting (or establishing)hereditary properties. This is usually done using (bounded/unbounded)approximate identities to obtain the extension as a limit of operatorsb->D(ba)-b.D(a), a in A, in an appropriate operator topology, themain point in the proof being to show that the limit map is in fact aderivation. In this paper we make clear which part of this approach isanalytic and which algebraic by presenting an algebraic scheme thatgives derivations in all situations at the cost of enlarging themodule. We use our construction to give improvements and shorterproofs of some resultsfrom the literatureand to give a necessary and sufficient condition that biprojectivity andbiflatness are inherited to ideals

M3 - Journal article

SN - 0253-4142

VL - 118

SP - 235

EP - 243

JO - Proceedings of the Indian Academy of sciences. Mathematical sciences

JF - Proceedings of the Indian Academy of sciences. Mathematical sciences

IS - 2

ER -

Push-outs of derivations

Abstract

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Citationsformater