Applications of the Banach-Stone Theorem on Algebra

For two compact Hausdorff spaces X and Y, C(X) and C(Y) are isomorphic if and only if X and Y are homeomorphic. This class result was given by Banach in 1932 and generalized by Stone in 1937. This opened the new way of research towards algebra isomerphisms. The algebraic version of Banach-Stone theorem asserts that C(X) and C(Y) are algebraic isomorphism if and only if X and Y are homeomerphic. In this paper we study the structure on the group of isometric isomorphism from C(X) to itself as an application of Banach-Stone theorem.