A class of (2m − 1)-weakly amenable Banach algebras

Let A be a Banach space and λ be a non-zero fixed element of A ∗ (dual space of A) with non-zero kernel. Defining algebra product in A as a · b = λ(a)b for a, b ∈ A, we show that A is a (2m− 1)-weakly amenable Banach algebra but not 2m-weakly amenable for any m ∈ N. Furthermore, we show the converse of the statement [2, Proposition 1.4.(ii)] “for a non-unital Banach algebra A, if A is weakly amenable then A# is weakly amenable” does not hold.