On the Lipschitz operator algebras. (English) Zbl 1211.47074

Ebadian, A.; Shokri, A. A.

Arch. Math., Brno 45, No. 3, 213-222 (2009).

Summary: In [Acta Math. Sin., Engl. Ser. 22, No. 3, 671–678 (2006; Zbl 1111.47045)], H. X. Cao, J. H. Zhang and Z. B. Xu defined an \(\alpha \)-Lipschitz operator from a compact metric space into a Banach space \(A\) and characterized it in the sense that \(F:K\rightarrow A\) is an \(\alpha \)-Lipschitz operator if and only if, for each \(\sigma \in X^{\ast }\), the mapping \(\sigma \circ F\) is an \(\alpha \)-Lipschitz function. The Lipschitz operator algebras \(L^{\alpha }(K,A)\) and \(l^{\alpha }(K,A)\) are developed here further, and we study the amenability and weak amenability of these algebras. Moreover, we prove the 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 \(L^{\alpha }_A(X,B)\).

MSC:

47B48	Linear operators on Banach algebras
46J10	Banach algebras of continuous functions, function algebras

Keywords:

Lipschitz algebra; amenability; homomorphism

Citations:

Zbl 1111.47045