Centralizing linear maps of an \(H^{*}\)-algebra

Let \(\mathcal {U}\) $\mathcal{U}$ be an algebra with center \(\mathcal {Z(U)}\) $\mathcal{Z}(\mathcal{U})$ . A mapping \(\phi :\mathcal {U}\rightarrow \mathcal {U}\) $\varphi :\mathcal{U}\to \mathcal{U}$ is centralizing if \(\phi (a)a-a\phi (a)\in \mathcal {Z(U)}\) $\varphi (a)a-a\varphi (a)\in \mathcal{Z}(\mathcal{U})$ for all \(a\in \mathcal {U}\) $a\in \mathcal{U}$ . We prove that any continuous centralizing linear map \(\phi \) $\varphi$ on a proper \(H^{*}\) $H_{}^{\ast }$ -algebra \(\mathcal {U}\) $\mathcal{U}$ with \(\mathcal {U}=\ell ^{2}(\Gamma , \mathcal {U}_{\gamma })\) $\mathcal{U}={\ell }^2(\mathrm{\Gamma },{\mathcal{U}}_{\gamma })$ ( each \(\mathcal {U}_{\gamma }\) ${\mathcal{U}}_{\gamma }$ is a minimal closed ideal of \(\mathcal {U}\) $\mathcal{U}$ ) is of the form \(\phi (a)=ca+\mu (a)\) $\varphi (a)=ca+\mu (a)$ , \(a\in \mathcal {U}\) $a\in \mathcal{U}$ , where \(c\in \ell ^{\infty }(\Gamma )\) $c\in {\ell }^{\infty }(\mathrm{\Gamma })$ and \(\mu :\mathcal {U}\rightarrow \mathcal {Z(U)}\) $\mu :\mathcal{U}\to \mathcal{Z}(\mathcal{U})$ is a continuous linear map. Then we examine the automatic continuity of centralizing linear maps on Banach algebras and by using it, a characterization of proper \(H^{*}\) $H_{}^{\ast }$ -algebras based on the automatic continuity of centralizing linear maps is given.