Let $(X,d)$ be a metric space and $Jsubseteq (0,infty)$ be
a nonempty set. We study the structure of the arbitrary intersection of
vector-valued Lipschitz algebras, and define a special Banach subalgebra of
$cap{Lip_gamma (X,E):gammain J}$, where $E$ is a Banach algebra, denoted by $ILip_J (X,E)$. Mainly,
we investigate $C-$character amenability of $ILip_J (X,E)$.
