Ring homomorphisms on commutative regular Banach algebras (Q950650)

Let \(A,B\) be commutative semisimple Banach algebras. A ring homomorphism \(\rho:A\to B\) is a mapping satisfying \(\rho(f+g)=\rho(f)+\rho(g)\) and \(\rho(fg)=\rho(f)\rho(g)\) for all \(f,g\in A\). Trivial ring homomorphisms on the ring of complex numbers \(\mathbb{C}\) are defined by \(\rho_{1}(z)=z\) and \(\rho_{-1}(z)=\overline{z}\) (complex conjugation) and \(\rho_{0}(z)=0\) for all \(z\in\mathbb{C}\). It is a well-known fact that there exist nontrivial ring homomorphisms on \(\mathbb{C}\). The main result of the paper is a partial representation of a ring homomorphism \(\rho:A\to B\) if it is assumed that \(A\) is in addition regular. Let \(A_{e}\) be the commutative Banach algebra obtained by adjoining a unit element \(e\) to \(A\). Let \(M_{A},M_{B}\) and \(M_{A_{e}}\) be the maximal ideal spaces of \(A,B\) and \(A_{e}\) respectively. Let \(M_{0}\) be the set of all \(y\in M_{B}\) such that \(y\circ\rho\equiv 0\). The main result states under the above-mentioned assumptions that there exist a decomposition of \(M_{B}\) into disjoint sets \(M_{0},M_{-1},M_{1}\) and \(M_{d}\) and a continuous mapping \(\varphi:M_{B}\setminus M_{0}\to M_{A_{e}}\) such that \(\rho(f)(y)\) is equal to \(0\) for all \(y\in M_{0},\) equal to \(f(\varphi(y))\) for all \(y\in M_{1},\) equal to \(\overline{f(\varphi(y))}\) for \(y\in M_{-1}\), and equal to \(\tau_{y}(q_{y}(f))\) for all \(y\in M_{d}\), where \(q_{y}\) is the quotient mapping from a prime ideal \(P_{y}\) of \(A\) onto \(A/P_{y}\) and \(\tau_{y}\) is a nonzero field homomorphism from the quotient field of \(P_{y}\) into \(\mathbb{C}\). As a corollary, the authors obtain the interesting result that there are no surjective ring homomorphism from \(C_{0}(\mathbb{R})\) onto \(C_{0}(\mathbb{D})\), where \(\mathbb{D}\) is the open unit disk in the complex plane.