The well known
Banach-Stone theorem states that if X and Y are locally compact Hausdorff spaces,
then the existence of an isometry φ of C0(Y ) onto C0(X) implies that X and Y are
homeomorphic. This result has been generalized by showing that the same
conclusion holds if the requirement that φ be an isometry is replaced by the
requirement that φ be an isomorphism with ∥φ∥∥φ−1∥ < 2. However, the author
knows of no valid examples in the literature which show that 2 is the largest
number for which this generalization is true. Here such an example is provided,
and it is shown that the reason for the apparent scarcity of examples is not
that they need be complicated, but rather, at least in the case where X is
compact and Y noncompact, that there is essentially just one way to construct
them.
