Homomorphisms between algebras of Lipschitz functions with the values in function algebras (Q530320)

The present paper aims at describing the algebraic homomorphisms between Banach algebras of Lipschitz mappings \(\mathrm{Lip}\big(X,C(M)\big)\), where \(X\) is a compact metric space, \(M\) is a compact Hausdorff topological space and the uniform algebra of complex-valued continuous functions \(C(M)\). Denote by \(\langle g,m \rangle=g(m)\) the action of \(g\in C(M)\) on \(m\in M\). The norm for \(f\in \mathrm{Lip}\big(X,C(M)\big)\) is defined by \(\|f\|_L=\sup_{x\in X}\|f(x)\|+L(f)\), where \(L(f)\) denotes the infimum of the Lispchitz constants of \(f\). The identity \(\mathbf{1}_X\) in \(\mathrm{Lip}\big(X,C(M)\big)\) is given by the constant mapping \(x\mapsto \mathbf{1}\). The algebraic homomorphisms \(\psi\) between two such algebras \(\mathrm{Lip}\big(X_1,C(M_1)\big)\) and \(\mathrm{Lip}\big(X_2,C(M_2)\big)\), with \(X_2\) connected, that preserve the identity are shown to be of the form \[ F\in \mathrm{Lip}\big(X_1,C(M_1)\big) \overset{\psi}{\longrightarrow}\big( y\mapsto {\ll} F \circ\delta_y\circ \varphi,\tau{\gg})\in \mathrm{Lip}\big(X_2,C(M_2)\big), \] where \(\tau:M_2\to M_1\) is a continuous mapping, \(\varphi:M_2\to \mathrm{Lip}(X_2,X_1)\) is a mapping with uniformly bounded Lipschitz constants, \(\delta_y\) denotes evaluation at the point \(y\in X_2\) and \({\ll},{\gg}\) denotes the pointwise action. In proving this result, the author also shows that the maximal ideal space of \(\mathrm{Lip}\big(X,C(M)\big)\) is homeomorphic to \(X\times M\). In the case when \(\psi\) is an algebraic isomorphism, then \(\tau\) is surjective and \(\delta_y\circ \varphi\) is continuous. When both \(X_1\) and \(X_2\) are connected, the algebraic isomorphisms are described in an analogous way to the above result.