Ideal amenability of module extensions of Banach algebras

Eshaghi M. Gordji; F. Habibian; B. Hayati

Archivum Mathematicum (2007)

  • Volume: 043, Issue: 3, page 177-184
  • ISSN: 0044-8753

Abstract

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Let 𝒜 be a Banach algebra. 𝒜 is called ideally amenable if for every closed ideal I of 𝒜 , the first cohomology group of 𝒜 with coefficients in I * is zero, i.e. H 1 ( 𝒜 , I * ) = { 0 } . Some examples show that ideal amenability is different from weak amenability and amenability. Also for n N , 𝒜 is called n -ideally amenable if for every closed ideal I of 𝒜 , H 1 ( 𝒜 , I ( n ) ) = { 0 } . In this paper we find the necessary and sufficient conditions for a module extension Banach algebra to be 2-ideally amenable.

How to cite

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Gordji, Eshaghi M., Habibian, F., and Hayati, B.. "Ideal amenability of module extensions of Banach algebras." Archivum Mathematicum 043.3 (2007): 177-184. <http://eudml.org/doc/250181>.

@article{Gordji2007,
abstract = {Let $\mathcal \{A\}$ be a Banach algebra. $\mathcal \{A\}$ is called ideally amenable if for every closed ideal $I$ of $\mathcal \{A\}$, the first cohomology group of $\mathcal \{A\}$ with coefficients in $I^*$ is zero, i.e. $H^1(\{\mathcal \{A\}\}, I^*)=\lbrace 0\rbrace $. Some examples show that ideal amenability is different from weak amenability and amenability. Also for $n\in \{N\}$, $\mathcal \{A\}$ is called $n$-ideally amenable if for every closed ideal $I$ of $\mathcal \{A\}$, $H^1(\{\mathcal \{A\}\},I^\{(n)\})=\lbrace 0\rbrace $. In this paper we find the necessary and sufficient conditions for a module extension Banach algebra to be 2-ideally amenable.},
author = {Gordji, Eshaghi M., Habibian, F., Hayati, B.},
journal = {Archivum Mathematicum},
keywords = {ideally amenable; Banach algebra; derivation; ideally amenable; Banach algebra; derivation},
language = {eng},
number = {3},
pages = {177-184},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Ideal amenability of module extensions of Banach algebras},
url = {http://eudml.org/doc/250181},
volume = {043},
year = {2007},
}

TY - JOUR
AU - Gordji, Eshaghi M.
AU - Habibian, F.
AU - Hayati, B.
TI - Ideal amenability of module extensions of Banach algebras
JO - Archivum Mathematicum
PY - 2007
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 043
IS - 3
SP - 177
EP - 184
AB - Let $\mathcal {A}$ be a Banach algebra. $\mathcal {A}$ is called ideally amenable if for every closed ideal $I$ of $\mathcal {A}$, the first cohomology group of $\mathcal {A}$ with coefficients in $I^*$ is zero, i.e. $H^1({\mathcal {A}}, I^*)=\lbrace 0\rbrace $. Some examples show that ideal amenability is different from weak amenability and amenability. Also for $n\in {N}$, $\mathcal {A}$ is called $n$-ideally amenable if for every closed ideal $I$ of $\mathcal {A}$, $H^1({\mathcal {A}},I^{(n)})=\lbrace 0\rbrace $. In this paper we find the necessary and sufficient conditions for a module extension Banach algebra to be 2-ideally amenable.
LA - eng
KW - ideally amenable; Banach algebra; derivation; ideally amenable; Banach algebra; derivation
UR - http://eudml.org/doc/250181
ER -

References

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  11. Haagerup U., All nuclear 𝒞 * -algebras are amenable, Invent. Math. 74 (1983), 305–319. (1983) MR0723220
  12. Johnson B. E., Cohomology in Banach algebras, Mem. Amer. Math. Soc. 127 (1972). (1972) Zbl0256.18014MR0374934
  13. Johnson B. E., Weak amenability of group algebras, Bull. London Math. Soc. 23 (1991), 281–284. (1991) Zbl0757.43002MR1123339
  14. Johnson B. E., White M. C., A non-weakly amenable augmentation ideal, submitted. 
  15. Wassermann S., On tensor products of certain group C * - algebras, J. Funct. Anal. 23 (1976), 28–36. (1976) Zbl0358.46040MR0425628
  16. Zhang, Yong, Weak amenability of module extensions of Banach algebras, Trans. Amer. Math. Soc. 354, (10) (2002), 4131–4151. Zbl1008.46019MR1926868

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