Generalized peripherally multiplicative maps between real Lipschitz algebras with involution

Document Type : Original Article


Department of Mathematics, Faculty of Science, Arak University



‎Let $(X,d)$ and $(Y,\rho)$ be compact metric spaces‎, ‎$\tau$ and $\eta$ be Lipschitz involutions on $ X$ and $Y$‎, ‎respectively‎, ‎$\mathcal{A}=Lip(X,d,\tau)$ and $\mathcal{B}=Lip(Y,\rho,\eta)$‎, ‎where $Lip(X,d,\tau)=\lbrace f\in Lip(X,d):f\circ\tau=\bar{f}\rbrace $‎. ‎For each $f\in \mathcal{A}$‎, ‎$\sigma_{\pi,\mathcal{A}}(f)$ denotes the peripheral spectrum of $f$‎. ‎We prove that if $S_{1},S_{2}:\mathcal{A}\rightarrow \mathcal{A}$ and $T_{1},T_{2}:\mathcal{A}\rightarrow \mathcal{B}$ are surjective mappings that satisfy $\sigma_{\pi,\mathcal{B}}(T_{1}(f)T_{2}(g))=\sigma_{\pi,\mathcal{A}}(S_{1}(f)S_{2}(g))$ for all $f,g\in \mathcal{A}$‎, ‎then there are $\kappa_{1},\kappa_{2}\in Lip(Y,\rho,\eta)$ with $\kappa_{1}\kappa_{2}=1_{Y}$ and a Lipschitz homeomorphism $\varphi$ from $(Y,\rho)$ to $(X,d)$ with $\tau \circ\varphi=\varphi \circ \eta$ on $Y$ such that $T_{j}(f)=\kappa_{j}\cdot(S_{j}(f)\circ\varphi)$ for all $f\in \mathcal{A}$ and $j=1,2$‎. ‎Moreover‎, ‎we show that the same result holds for surjective mappings $S_{1},S_{2}:\mathcal{A}\rightarrow \mathcal{A}$ and $T_{1},T_{2}:\mathcal{A}\rightarrow \mathcal{B}$ that satisfy $\sigma_{\pi,\mathcal{B}}(T_{1}(f)T_{2}(g))\cap\sigma_{\pi,\mathcal{A}}(S_{1}(f)S_{2}(g))\neq\emptyset$ for all $f,g\in \mathcal{A}$‎.