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On the Lipschitz operator algebras. (English) Zbl 1211.47074

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

Citations:

Zbl 1111.47045
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