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 $\mathrm C^*$-algebra onto another commutative $\mathrm C^*$-algebra with connected infinite Gelfand space is either linear or anti-linear.

(ii)

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

(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 $L^p(G)$ $(1\le p<\infty, G \text{a compact abelian group})$.