ABSTRACT
If two real function algebras are linearly isometric (as Banach spaces), can we say that those algebras are isomorphic (as algebras)? This is the central question of the present chapter. We shall answer this question in the affirmative. The origin of such questions can be traced back to the classical Banach-Stone theorem, which states that if C(X) and C(Y) are linearly isometric, where X and Y are compact Hausdorff spaces, then X and Y are homeomorphic. This implies that C(X) and C(Y) are isomorphic. Several proofs of this theorem are available in the literature. We refer the reader to the surveys by Behrends (1979) and Jarosz (1985) for these proofs as well as for many generalizations of the Banach-Stone theorem. In this chapter we shall be concerned with one such generalization due to Nagasawa (1959). He proved that if two complex function algebras are linearly isometric, then those algebras are isomorphic. Our aim is to extend this result to the case of real function algebras.
