# Article

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Keywords:
Lipschitz algebras; amenability; homomorphism
Summary:
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 $L^\alpha _A(X,B)$.
References:
[1] Alimohammadi, D., Ebadian, A.: Headberg’s theorem in real Lipschitz algebras. Indian J. Pure Appl. Math. 32 (2001), 1479–1493. MR 1878062
[2] Bade, W. G., Curtis, P. C., Dales, H. G.: Amenability and weak amenability for Berurling and Lipschitz algebras. Proc. London Math. Soc. 55 (3) (1987), 359–377. MR 0896225
[3] Cao, H. X., Xu, Z. B.: Some properties of Lipschitz-$\alpha$ operators. Acta Math. Sin. (Engl. Ser.) 45 (2) (2002), 279–286. MR 1928136
[4] Cao, H. X., Zhang, J. H., Xu, Z. B.: Characterizations and extentions of Lipschitz-$\alpha$ operators. Acta Math. Sin. (Engl. Ser.) 22 (3) (2006), 671–678. DOI 10.1007/s10114-005-0727-x | MR 2219676
[5] Dales, H. G.: Banach Algebras and Automatic Continuty. Clarendon Press, Oxford, 2000. MR 1816726
[6] Ebadian, A.: Prime ideals in Lipschitz algebras of finite differentable function. Honam Math. J. 22 (2000), 21–30. MR 1779197
[7] Honary, T. G, Mahyar, H.: Approximation in Lipschitz algebras. Quaest. Math. 23 (2000), 13–19. DOI 10.2989/16073600009485953 | MR 1796246 | Zbl 0963.46034
[8] Johnson, B. E.: Cohomology in Banach algebras. Mem. Amer. Math. Soc. 127 (1972). MR 0374934 | Zbl 0256.18014
[9] Johnson, B. E.: Lipschitz spaces. Pacific J. Math. 51 (1975), 177–186. DOI 10.2140/pjm.1974.51.177 | MR 0346503
[10] Runde, V.: Lectures on Amenability. Springer, 2001. MR 1874893
[11] Sherbert, D. R.: Banach algebras of Lipschitz functions. Pacific J. Math. 3 (1963), 1387–1399. DOI 10.2140/pjm.1963.13.1387 | MR 0156214 | Zbl 0121.10203
[12] Sherbert, D. R.: The structure of ideals and point derivations in Banach algebras of Lipschitz functions. Trans. Amer. Math. Soc. 111 (1964), 240–272. DOI 10.1090/S0002-9947-1964-0161177-1 | MR 0161177 | Zbl 0121.10204
[13] Weaver, N.: Subalgebras of little Lipschitz algebras. Pacific J. Math. 173 (1996), 283–293. MR 1387803 | Zbl 0846.54013
[14] Weaver, N.: Lipschitz Algebras. World Scientific Publishing Co., Inc., River Edge, NJ, 1999. MR 1832645 | Zbl 0936.46002

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