## Automatic continuity of certain isomorphisms between regular Banach function algebras.(English)Zbl 0901.46042

If $$A$$ and $$B$$ are regular commutative semisimple Banach algebras then a linear map $$T:A\to B$$ is said to be separating (or disjointness preserving) if $$fg=0$$ implies $$TfTg=0$$. Here it is shown that if $$A$$ satisfies Ditkin’s condition then a separating bijection is necessarily continuous and its inverse is separating. If $$B$$ also satisfies Ditkin’s condition then the structure spaces of the two algebras are homeomorphic. In particular, it is shown that linear isometries between regular uniform algebras are separating; classical results, like the Banach-Stone theorem, follow.

### MSC:

 46H40 Automatic continuity
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### References:

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