The BSE concepts for vector-valued Lipschitz algebras

Abstract

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ (K,d) $\end{document}</tex-math></inline-formula> be a compact metric space, <inline-formula><tex-math id="M2">\begin{document}$ \mathcal A $\end{document}</tex-math></inline-formula> be a commutative semisimple Banach algebra and <inline-formula><tex-math id="M3">\begin{document}$ 0&lt;\alpha\leq 1 $\end{document}</tex-math></inline-formula>. The overall purpose of the present paper is to demonstrate that all BSE concepts of <inline-formula><tex-math id="M4">\begin{document}$ {\rm Lip}_\alpha(K,\mathcal A) $\end{document}</tex-math></inline-formula> are inherited from <inline-formula><tex-math id="M5">\begin{document}$ \mathcal A $\end{document}</tex-math></inline-formula> and vice versa. Recently, the authors proved in the case that <inline-formula><tex-math id="M6">\begin{document}$ \mathcal A $\end{document}</tex-math></inline-formula> is unital, <inline-formula><tex-math id="M7">\begin{document}$ {\rm Lip}_\alpha(K,\mathcal A) $\end{document}</tex-math></inline-formula> is a BSE-algebra if and only if <inline-formula><tex-math id="M8">\begin{document}$ \mathcal A $\end{document}</tex-math></inline-formula> is so. In this paper, we generalize this result for an arbitrary commutative semisimple Banach algebra <inline-formula><tex-math id="M9">\begin{document}$ \mathcal A $\end{document}</tex-math></inline-formula>. Furthermore, we investigate the BSE-norm property for <inline-formula><tex-math id="M10">\begin{document}$ {\rm Lip}_\alpha(K,\mathcal A) $\end{document}</tex-math></inline-formula> and prove that <inline-formula><tex-math id="M11">\begin{document}$ {\rm Lip}_\alpha(K,\mathcal A) $\end{document}</tex-math></inline-formula> belongs to the class of BSE-norm algebras if and only if <inline-formula><tex-math id="M12">\begin{document}$ \mathcal A $\end{document}</tex-math></inline-formula> is owned by this class. Moreover, we prove that for any natural number <inline-formula><tex-math id="M13">\begin{document}$ n $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M14">\begin{document}$ n\geq 2 $\end{document}</tex-math></inline-formula>, if all continuous bounded functions on <inline-formula><tex-math id="M15">\begin{document}$ \Delta({\rm Lip}_\alpha(K,\mathcal A)) $\end{document}</tex-math></inline-formula> are <inline-formula><tex-math id="M16">\begin{document}$ n $\end{document}</tex-math></inline-formula>-BSE-functions, then <inline-formula><tex-math id="M17">\begin{document}$ K $\end{document}</tex-math></inline-formula> is finite. As a result, we obtain that <inline-formula><tex-math id="M18">\begin{document}$ {\rm Lip}_{\alpha}(K,\mathcal A) $\end{document}</tex-math></inline-formula> is a BSE-algebra of type I if and only if <inline-formula><tex-math id="M19">\begin{document}$ \mathcal A $\end{document}</tex-math></inline-formula> is a BSE-algebra of type I and <inline-formula><tex-math id="M20">\begin{document}$ K $\end{document}</tex-math></inline-formula> is finite. Furthermore, in according to a result of Kaniuth and Ülger, which disapproves the BSE-property for <inline-formula><tex-math id="M21">\begin{document}$ {\rm lip}_{\alpha}K $\end{document}</tex-math></inline-formula>, we show that for any commutative semisimple Banach algebra <inline-formula><tex-math id="M22">\begin{document}$ \mathcal A $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M23">\begin{document}$ {\rm lip}_{\alpha}(K,\mathcal A) $\end{document}</tex-math></inline-formula> fails to be a BSE-algebra, as well. Finally, we concentrate on the classical Lipschitz algebra <inline-formula><tex-math id="M24">\begin{document}$ {\rm Lip}_\alpha X $\end{document}</tex-math></inline-formula>, for an arbitrary metric space (not necessarily compact) <inline-formula><tex-math id="M25">\begin{document}$ (X,d) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M26">\begin{document}$ \alpha&gt;0 $\end{document}</tex-math></inline-formula>, when <inline-formula><tex-math id="M27">\begin{document}$ {\rm Lip}_\alpha X $\end{document}</tex-math></inline-formula> separates the points of <inline-formula><tex-math id="M28">\begin{document}$ X $\end{document}</tex-math></inline-formula>. In particular, we show that <inline-formula><tex-math id="M29">\begin{document}$ {\rm Lip}_\alpha X $\end{document}</tex-math></inline-formula> is a BSE-algebra, as well as a BSE-norm algebra.