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Article Contents

# The BSE concepts for vector-valued Lipschitz algebras

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• Let $(K,d)$ be a compact metric space, $\mathcal A$ be a commutative semisimple Banach algebra and $0<\alpha\leq 1$. The overall purpose of the present paper is to demonstrate that all BSE concepts of ${\rm Lip}_\alpha(K,\mathcal A)$ are inherited from $\mathcal A$ and vice versa. Recently, the authors proved in the case that $\mathcal A$ is unital, ${\rm Lip}_\alpha(K,\mathcal A)$ is a BSE-algebra if and only if $\mathcal A$ is so. In this paper, we generalize this result for an arbitrary commutative semisimple Banach algebra $\mathcal A$. Furthermore, we investigate the BSE-norm property for ${\rm Lip}_\alpha(K,\mathcal A)$ and prove that ${\rm Lip}_\alpha(K,\mathcal A)$ belongs to the class of BSE-norm algebras if and only if $\mathcal A$ is owned by this class. Moreover, we prove that for any natural number $n$ with $n\geq 2$, if all continuous bounded functions on $\Delta({\rm Lip}_\alpha(K,\mathcal A))$ are $n$-BSE-functions, then $K$ is finite. As a result, we obtain that ${\rm Lip}_{\alpha}(K,\mathcal A)$ is a BSE-algebra of type I if and only if $\mathcal A$ is a BSE-algebra of type I and $K$ is finite. Furthermore, in according to a result of Kaniuth and Ülger, which disapproves the BSE-property for ${\rm lip}_{\alpha}K$, we show that for any commutative semisimple Banach algebra $\mathcal A$, ${\rm lip}_{\alpha}(K,\mathcal A)$ fails to be a BSE-algebra, as well. Finally, we concentrate on the classical Lipschitz algebra ${\rm Lip}_\alpha X$, for an arbitrary metric space (not necessarily compact) $(X,d)$ and $\alpha>0$, when ${\rm Lip}_\alpha X$ separates the points of $X$. In particular, we show that ${\rm Lip}_\alpha X$ is a BSE-algebra, as well as a BSE-norm algebra.

Mathematics Subject Classification: Primary:46J05;Secondary:46J10.

 Citation:

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