On homological notions of Banach algebras related to a character

Abstract

In this paper, we countinue our work in \cite{11}. We show that $L^1(G,w)$ is ${\varphi }_0$-biprojective if and only if $G$ is compact, where ${\varphi }_0$ is the augmentation character. We introduce the notions of character Johnson amenability and character Johnson contractibility for Banach algebras. We show that ${\ell }^1(S)$ is pseudo-amenable if and only if ${\ell }^1(S)$ is character Johnson-amenable, provided that $S$ is a uniformly locally finite band semigroup. We give some conditions whether $\varphi$-biprojectivity ($\varphi$-biflatness) of ${\ell }^1(S)$ implies the finiteness (amenability) of $S$, respectively.

arXiv comment10 pages