The Bochner–Eberlein–Doss property for $$\textrm{Lip}(X,{\mathcal {A}})$$

Let (X, d) be a compact metric space and \({\mathcal {A}}\) be a commutative and semisimple Banach algebra. Some of our recent works are related to the several \(\mathrm BSE\) concepts of the vector-valued Lipschitz algebra \(\textrm{Lip}(X,{\mathcal {A}})\). In this paper as the main purpose, we verify the \(\mathrm BED\) property for \(\textrm{Lip}(X,{\mathcal {A}})\), which is actually different from the \(\mathrm BSE\) feature. We first prove as an elementary result that \(\textrm{Lip}(X,{\mathcal {A}})\) is regular if and only if \({\mathcal {A}}\) is so. Then we prove that \({\mathcal {A}}\) is a \(\mathrm BED\) algebra, whenever \(\textrm{Lip}(X,{\mathcal {A}})\) is so. Afterwards, we verify the converse of this statement. Indeed, we prove that if \({\mathcal {A}}\) is a \(\mathrm BED\) algebra then \(C^{0}_{\textrm{BSE}}(\Delta (\textrm{Lip}(X,{\mathcal {A}})))\subseteq \widehat{\textrm{Lip}(X,{\mathcal {A}})}\) and \(\widehat{\textrm{Lip}X\otimes {\mathcal {A}}}\subseteq C^{0}_{\textrm{BSE}}(\Delta (\textrm{Lip}(X,{\mathcal {A}}))).\) It follows that if \(\textrm{Lip}X\otimes {\mathcal {A}}\) is dense in \(\textrm{Lip}(X,{\mathcal {A}})\) then \(\textrm{Lip}(X,{\mathcal {A}})\) is a \(\mathrm BED\) algebra, provided that \({\mathcal {A}}\) is so. Moreover, we conclude that the necessary and sufficient condition for the unital and in particular finite dimensional Banach algebra \({\mathcal {A}}\), to be a \(\mathrm BED\) algebra is that \(\textrm{Lip}(X,{\mathcal {A}})\) is a \(\mathrm BED\) algebra. Finally, regarding to some known results which disapproves the \(\mathrm BSE\) property for \(\textrm{lip}_{\alpha }(X,{\mathcal {A}})\) \((0<\alpha <1\)), we show that for any commutative and semisimple Banach algebra \({\mathcal {A}}\) with \({{\mathcal {A}}}_0\ne \emptyset \), \(\textrm{lip}_{\alpha }(X,{\mathcal {A}})\) fails to be a \(\mathrm BED\) algebra, as well.