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Davood Alimohammadi

Davood Alimohammadi

Academic rank: Associate Professor
ORCID: https://orcid.org/0000-0002-9398-6213
Education: PhD.
ScopusId: 6505995626
HIndex:
Faculty: Science
Address: Arak University
Phone:

Research

Title
Closed Ideals, Point Derivations and Weak Amenability of Extended Little Lipschitz Algebras
Type
JournalPaper
Keywords
Banach function algebra, Extended Lipschitz algebra, Point derivation, Weak amenability
Year
2016
Journal Caspian Journal of Mathematical Sciences
DOI
Researchers MALIHEH MAYGHANI ، Davood Alimohammadi

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 ${\rm Lip}(X,K,d^ \alpha)$ 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 equipped the algebra norm $||f||_{{\rm Lip}(X, K, d^ {\alpha})}= ||f||_X+ p_{(K,d^{\alpha})}(f)$, where $||f||_X=\sup\{|f(x)|:~x\in X \}$. We denote by ${\rm lip}(X,K,d^ \alpha )$ the closed subalgebra of ${\rm Lip}(X,K,d^ \alpha)$ consisting of all $f\in {\rm Lip}(X,K,d^ \alpha)$ for which $\frac{|f(x)-f(y)|}{d^\alpha(x,y)}\to 0$ as $d(x, y)\to 0$ with $x, y \in K$. In this paper we show that every proper closed ideal of $({\rm lip}(X,K,d^\alpha),\|\cdot\|_{Lip(X,K,d^\alpha)})$ is the intersection of all maximal ideals containing it. We also prove that every continuous point derivation of ${\rm lip}(X,K,d^\alpha)$ is zero. Next we show that ${\rm lip}(X,K,d^\alpha)$ is weakly amenable if $\alpha \in (0, \frac{1} {2})$. We also prove that ${\rm lip}(\Bbb {T},K,d^{\frac{1}{2}})$ is weakly amenable where $\Bbb {T}=\{z\in\Bbb C : |z|=1\}$, $d$ is the Euclidean metric on $\Bbb T$ and $K$ is a nonempty compact set in $(\Bbb T,d)$.