In a recent paper by H. X. Cao, J. H. Zhang and Z. B. Xu an $\alpha$-Lipschitz operator from a compact metric space into a Banach space $A$ is defined and characterized in a natural way in the sence that $F:K\to A$ is a $\alpha$-Lipschitz operator if and only if for each $\sigma \in X^{*}$ the mapping $\sigma \circ F$ is a $\alpha$-Lipschitz function. The Lipschitz operators algebras $L^{\alpha }(K,A)$ and $l^{\alpha }(K,A)$ are developed here further, and we study their amenability and weak amenability of these algebras. Moreover, we prove an interesting result that $L^{\alpha }(K,A)$ and $l^{\alpha }(K,A)$ are isometrically...

We prove that, for certain domains $\mathbb{D}$, continuous product of domains $D_{\omega }$, the Carathéodory pseudodistance satisfies the following product property $C_{\mathbb{D}}\left(f,g\right)={sup}_{\omega }{C}_{D_{\omega }}\left(f\left(\omega \right),g\left(\omega \right)\right)$

For any uniformly closed subalgebra A of C(K) for a compact Hausdorff space K without isolated points and $x_0\in A$, we show that every complete norm on A which makes continuous the multiplication by $x_0$ is equivalent to $\parallel ·{\parallel }_{\infty }$ provided that $x_0^{-1}\left(\lambda \right)$ has no interior points whenever λ lies in ℂ. Actually, these assertions are equivalent if A = C(K).