## Bulletin of the Belgian Mathematical Society - Simon Stevin

### On $\phi$-biflat and $\phi$-biprojective Banach algebras

#### Abstract

In this paper, we introduce the new notions of $\phi$-biflatness, $\phi$-biprojectivity, $\phi$-Johnson amenability and $\phi$-Johnson contractibility for Banach algebras, where $\phi$ is a non-zero homomorphism from a Banach algebra $A$ into $\mathbb{C}$. We show that a Banach algebra $A$ is $\phi$-Johnson amenable if and only if it is $\phi$-inner amenable and $\phi$-biflat. Also we show that $\phi$-Johnson amenability is equivalent with the existence of left and right $\phi$-means for $A$. We give some examples to show differences between these new notions and the classical ones. Finally, we show that ${L^{1}(G)}$ is $\phi$-biflat if and only if $G$ is an amenable group and $A(G)$ is $\phi$-biprojective if and only if $G$ is a discrete group.

#### Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 20, Number 5 (2013), 789-801.

Dates
First available in Project Euclid: 25 November 2013

https://projecteuclid.org/euclid.bbms/1385390764

Digital Object Identifier
doi:10.36045/bbms/1385390764

Mathematical Reviews number (MathSciNet)
MR3160589

Zentralblatt MATH identifier
1282.43001

#### Citation

Sahami, A.; Pourabbas, A. On $\phi$-biflat and $\phi$-biprojective Banach algebras. Bull. Belg. Math. Soc. Simon Stevin 20 (2013), no. 5, 789--801. doi:10.36045/bbms/1385390764. https://projecteuclid.org/euclid.bbms/1385390764