are regular commutative semisimple Banach algebras then a linear map
is said to be separating (or disjointness preserving) if
. Here it is shown that if
satisfies Ditkin’s condition then a separating bijection is necessarily continuous and its inverse is separating. If
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.