A Structure of Ring Homomorphisms on Commutative Banach Algebras

We give a structure theorem for a ring homomorphism of a commutative regular Banach algebra into another commutative Banach algebra. In particular, it is shown that: (i) A ring homomorphism of a commutative C∗-algebra onto another commutative C∗-algebra with connected infinite Gelfand space is either linear or anti-linear. (ii) A ring automorphism of L1(R ) is either linear or anti-linear. (iii) Cn([a, b]), L1(R ) and the disc algebra A(D) are neither ring homomorphic images of `1(S) nor Lp(G) (1 ≤ p <∞, G a compact abelian group). Let A and B be two commutative Banach algebras with Gelfand spaces ΦA and ΦB, respectively. Let ρ be a ring homomorphism of A into B such that {ρ(x)̂(ψ) : x ∈ A} = C, the complex field, (∗) for each ψ ∈ ΦB (“̂” denotes the Gelfand transform). This, of course, holds if ρ is onto. The purpose of this note is to show the following structure theorem of ρ applying the method which L. Molnar used in [5] to prove that a commutative semisimple Banach algebra which is the range of a ring homomorphism from a commutative C∗-algebra must be C∗-equivalent. Theorem 1. Suppose A is regular. Then there exist a continuous map ρ̂ of ΦB into ΦA and a division {ΦB,ΦB,ΦB} of ΦB such that ΦB and ΦB are closed, and for each a ∈ A, ρ(a)̂ = â ◦ ρ̂ on ΦB, ρ(a)̂ = ̄̂ a ◦ ρ̂ on ΦB and ρ(a)̂(ψ) = τψ(â(ρ̂(ψ))) for every ψ ∈ ΦB and for a certain discontinuous ring automorphism τψ of the complex field C. Moreover, if ρ is surjective, then ρ̂ is injective, and if A satisfies the following condition (#), then ρ̂(ΦB) is a finite set: (#) For any λn∈C with |λn|≤1/2n (n = 1, 2, . . . ) and any sequence {φ1, φ2, . . . } in ΦA such that each φn is an isolated point in {φ1, φ2, . . . }, there exists an element a ∈ A such that â(φn) = λn (n = 1, 2, . . . ). Received by the editors May 29, 1997 and, in revised form, October 27, 1997. 1991 Mathematics Subject Classification. Primary 46J05, 46E25.