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

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