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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.

46H40Automatic continuity
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