Cohomological properties of different types of weak amenability

Abstract

In this paper, we deal with cohomological properties of weak amenability, cyclic amenability, cyclic weak amenability and point amenability of Banach algebras. We look at some hereditary properties of them and show that continuous homomorphisms with dense range preserve cyclically weak amenability, however, weak amenability and cyclically amenability are preserved under certain conditions. We also study these cohomological properties of the $\theta-$Lau product $A\times_\theta B$ and the projective tensor product $A\hat{\otimes} B$. Finally, we investigate the cohomological properties of $A^{**}$ and establish that cyclically weak amenability of $A^{**}$ implies cyclically weak amenability of $A$. This result is true for point amenability instead of cyclically weak amenability.