Cohomological properties of different types of weak amenability

TL;DR

This paper investigates various cohomological properties related to weak amenability in Banach algebras, exploring their hereditary nature and behavior under algebraic constructions like homomorphisms, products, and duals.

Contribution

It provides new insights into how cohomological properties are preserved or transferred in Banach algebras and their duals, especially under specific algebraic operations.

Findings

Continuous homomorphisms with dense range preserve cyclically weak amenability.

Weak amenability and cyclically amenability are preserved under certain conditions.

Cyclically weak amenability of the second dual implies cyclically weak amenability of the original algebra.

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.