A structure of ring homomorphisms on commutative Banach algebras

Abstract:

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:

1. [(i)] A ring homomorphism of a commutative $\mathrm C^*$-algebra onto another commutative $\mathrm C^*$-algebra with connected infinite Gelfand space is either linear or anti-linear.

2. [(ii)] A ring automorphism of $L^1(\boldsymbol {R}^N)$ is either linear or anti-linear.

3. [(iii)] $C^n([a,b])$, $L^1(\boldsymbol {R}^N)$ and the disc algebra $A(D)$ are neither ring homomorphic images of $\ell ^1(S)$ nor $L^p(G)$ $(1\le p<\infty , G \text {a compact abelian group})$.