On isomorphisms between ideals in rings of continuous functions

Abstract:

A ring of continuous functions is a ring of the form $C(X)$, the ring of all continuous real-valued functions on a completely regular Hausdorff space $X$. For an arbitrary ideal $I$ in $C(X)$, the author shows that the maximal ideals of $I$ are precisely the ideals of the form $I \cap M$, for some maximal ideal $M$ in $C(X)$ not containing $I$. The author shows that any ring isomorphism between ideals in any two rings of continuous functions preserves order, boundedness, and lattice structure; and he uses these results to obtain one of the main theorems: An isomorphism of a maximal ideal in $C(X)$ onto a maximal idea in $C(Y)$ can be extended to an isomorphism of $C(X)$ onto $C(Y)$. Another of the main theorems characterizes those isomorphisms between ${C^ \ast }(X)$ and ${C^ \ast }(Y)$ (the subrings of bounded functions in $C(X)$ and $C(Y)$ respectively) which can be extended to isomorphisms between $C(X)$ and $C(Y)$. The author proves that, given any ideal $I$ in $C(X)$, there exists a space $X(I)$ so that the uniform closure of $I$ is isomorphic to a maximal ideal in $C(X(I))$.