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

A. Sahami¹ ; A. Pourabbas²

¹ Department 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
² Faculty of Mathematics and Computer Science Amirkabir University of Technology 424 Hafez Avenue 15914 Tehran, Iran

Colloquium Mathematicum, Tome 145 (2016) no. 2, pp. 273-284

Résumé

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 $G$.

Détail

DOI : 10.4064/cm6459-11-2015

Keywords: first part paper investigate approximate biprojectivity banach algebras related locally compact groups segal algebra approximate biprojective only compact every continuous weight approximate biprojective only compact provided geq every second part study phi biflatness banach algebras where phi character phi biflat amenable group where phi augmentation character finally phi biflatness ** implies amenability nbsp