Document Type : Research Paper

Authors

• Maliheh Mayghani ¹
• Davood Alimohammadi ²

¹ Department of Mathematics, Payame Noor University, Tehran, 19359-3697, Iran

² Department of Mathematics, Faculty of Science, Arak University, Arak, Iran

Abstract

Let $(X,d)$ be a compact
metric space and let $K$ be a nonempty compact subset of $X$. Let $\alpha \in (0,1]$ and let $\mathrm{L}\mathrm{i}\mathrm{p}(X,K,d^{\alpha })$ denote the Banach algebra of all  continuous complex-valued functions $f$ on
$X$ for which
$$p_{(K,d^{\alpha })}(f)=sup\{\frac{|f(x)-f(y)|}{d^{\alpha }(x,y)}:x,y\in K,x\ne y\}<\mathrm{\infty }$$
when it is equipped with the algebra norm $||f|{|}_{\mathrm{L}\mathrm{i}\mathrm{p}(X,K,d^{\alpha })}=||f|{|}_X+p_{(K,d^{\alpha })}(f)$, where $||f|{|}_X=sup\{|f(x)|:x\in X\}$.  In this paper we first study the structure of certain ideals of $\mathrm{L}\mathrm{i}\mathrm{p}(X,K,d^{\alpha })$. Next we show that if $K$  is infinite and $\mathrm{i}\mathrm{n}\mathrm{t}(K)$ contains a limit point of $K$ then $\mathrm{L}\mathrm{i}\mathrm{p}(X,K,d^{\alpha })$ has at least a nonzero continuous point derivation and applying this fact we prove that $\mathrm{L}\mathrm{i}\mathrm{p}(X,K,d^{\alpha })$ is not weakly amenable and amenable.

Keywords

• Amenability
• Banach function algebra
• extended Lipschitz algebra
• point derivation
• weak amenability