# On the Lipschitz operator algebras

## A. Ebadian and A. A. Shokri

Address:

Faculty of Basic Science, Science and Research Brunch Islamic Azad University (IAU), Tehran, Iran

Department of Mathematics, Faculty of Science Urmia University, Urmia, Iran

E-mail:

a-shokri@iau-ahar.ac.ir

a.ebadian@urmia.ac.ir

Abstract: 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 $F:K\rightarrow A$ 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 isomorphic to $L^{\alpha }(K)\check{\otimes }A$ and $l^{\alpha }(K)\check{\otimes }A$ respectively. Also we study homomorphisms on the $L^\alpha _A(X,B)$.

AMSclassification: primary 47B48; secondary 46J10.

Keywords: Lipschitz algebras, amenability, homomorphism.