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Let 
$(X,d)$ be a metric space and let 
$J\subseteq [0,\infty )$ be nonempty. We study the structure of the arbitrary intersections of Lipschitz algebras and define a special Banach subalgebra of 
${{\cap }_{\gamma \in J\,}}\text{Li}{{\text{p}}_{\gamma \,}}X$, denoted by 
$\text{ILi}{{\text{p}}_{J}}X$. Mainly, we investigate the 
$C$-character amenability of $\text{ILi}{{\text{p}}_{J}}X$, in particular Lipschitz algebras. We address a gap in the proof of a recent result in this field. Then we remove this gap and obtain a necessary and sufficient condition for $C$-character amenability of $\text{ILi}{{\text{p}}_{J}}X$, specially Lipschitz algebras, under an additional assumption.
Let 
${\it\varphi}$ be a homomorphism from a Banach algebra 
${\mathcal{B}}$ to a Banach algebra 
${\mathcal{A}}$. We define a multiplication on the Cartesian product space 
${\mathcal{A}}\times {\mathcal{B}}$ and obtain a new Banach algebra 
${\mathcal{A}}\times _{{\it\varphi}}{\mathcal{B}}$. We show that biprojectivity as well as biflatness of 
${\mathcal{A}}\times _{{\it\varphi}}{\mathcal{B}}$ are stable with respect to 
${\it\varphi}$.
