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Approximate biprojectivity and $\phi $-biflatness of certain Banach algebras

Volume 145 / 2016

A. Sahami, A. Pourabbas Colloquium Mathematicum 145 (2016), 273-284 MSC: Primary 43A07, 43A20; Secondary 46H05. DOI: 10.4064/cm6459-11-2015 Published online: 8 July 2016

Abstract

In the first part of the paper, we investigate the approximate biprojectivity of some Banach algebras related to the locally compact groups. We show that a Segal algebra $S(G)$ is approximate biprojective if and only if $G$ is compact. Also for every continuous weight $w$, we show that $L^{1}(G,w)$ is approximate biprojective if and only if $G$ is compact, provided that $w(g)\geq 1$ for every $g\in G$.

In the second part, we study $\phi $-biflatness of some Banach algebras, where $\phi $ is a character. We show that if $S(G)$ is $\phi _{0}$-biflat, then $G$ is an amenable group, where $\phi _{0}$ is the augmentation character on $S(G)$. Finally, we show that the $\phi $-biflatness of $L^{1}(G)^{**}$ implies the amenability of $G$.

Authors

  • A. SahamiDepartment of Mathematics
    Faculty of Basic Sciences
    Ilam University
    P.O. Box 69315-516
    Ilam, Iran
    and
    Faculty of Mathematics and Computer Science
    Amirkabir University of Technology
    424 Hafez Avenue
    15914 Tehran, Iran
    e-mail
  • A. PourabbasFaculty of Mathematics and Computer Science
    Amirkabir University of Technology
    424 Hafez Avenue
    15914 Tehran, Iran
    e-mail

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