Vol. 35, No. 2, 1970

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Isomorphisms of C0(Y ) onto C(X)

Michael James Cambern

Vol. 35 (1970), No. 2, 307–312

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∥∥φ1< 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.

Mathematical Subject Classification 2000
Primary: 46E10
Secondary: 54C10
Received: 12 June 1969
Published: 1 November 1970
Michael James Cambern