We show that a free graded commutative Banach algebra over a (purely odd) Banach space $E$ is a Banach-Grassmann algebra in the sense of Jadczyk and Pilch if and only if $E$ is infinite-dimensional. Thus, a large amount of new examples of separable Banach-Grassmann algebras arise in addition to the only one example previously known due to A. Rogers.

The generalized notion of weak amenability, namely $(\varphi ,\psi )$-weak amenability, where $\varphi ,\psi$ are continuous homomorphisms on a Banach algebra $𝒜$, was introduced by Bodaghi, Eshaghi Gordji and Medghalchi (2009). In this paper, the $(\varphi ,\psi )$-weak amenability on the measure algebra $M\left(G\right)$, the group algebra $L^1\left(G\right)$ and the Segal algebra $S^1\left(G\right)$, where $G$ is a locally compact group, are studied. As a typical example, the $(\varphi ,\psi )$-weak amenability of a special semigroup algebra is shown as well.