The structure of ideals, point derivations, amenability and weak amenability of extended Lipschitz algebras | ||
| International Journal of Nonlinear Analysis and Applications | ||
| مقاله 30، دوره 8، شماره 1، 2017، صفحه 389-404 اصل مقاله (438.09 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22075/ijnaa.2016.493 | ||
| نویسندگان | ||
| Maliheh Mayghani1؛ Davood Alimohammadi* 2 | ||
| 1Department of Mathematics, Payame Noor University, Tehran, 19359-3697, Iran | ||
| 2Department of Mathematics, Faculty of Science, Arak University, Arak, Iran | ||
| چکیده | ||
| 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 ${\rm Lip}(X,K,d^ \alpha)$ denote the Banach algebra of all continuous complex-valued functions $f$ on $X$ for which $$p_{(K,d^\alpha)}(f)=\sup\{\frac{|f(x)-f(y)|}{d^\alpha(x,y)} : x,y\in K , x\neq y\}<\infty$$ when it is equipped with the algebra norm $||f||_{{\rm Lip}(X, K, d^ {\alpha})}= ||f||_X+ p_{(K,d^{\alpha})}(f)$, where $||f||_X=\sup\{|f(x)|:~x\in X \}$. In this paper we first study the structure of certain ideals of ${\rm Lip}(X,K,d^\alpha)$. Next we show that if $K$ is infinite and ${\rm int}(K)$ contains a limit point of $K$ then ${\rm Lip}(X,K,d^\alpha)$ has at least a nonzero continuous point derivation and applying this fact we prove that ${\rm Lip}(X,K,d^\alpha)$ is not weakly amenable and amenable. | ||
| کلیدواژهها | ||
| Amenability؛ Banach function algebra؛ extended Lipschitz algebra؛ point derivation؛ weak amenability | ||
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