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LetAbe a normal Fréchet function algebra on a topological spaceXwhich is the projective limit of a sequence of certain Banach function algebras and satisfies Ditkin's condition. LetBbe a function algebra on a topological spaceY. We show that ifT:A → Bis a separating map, that isf.g= 0 impliesT f.T g= 0, thenTcan be represented asT f(y) =T1(y)f(h(y)), for eachf∊A, on a subset ofY. Using this result we show that ifBis also a Fréchet function algebra andTis a separating bijection then it is automatically continuous. [ABSTRACT FROM AUTHOR]

Let A and B be strongly regular normal Fréchet function algebras on compact Hausdorff spaces X and Y, respectively, such that the evaluation homomorphisms are continuous on A and B. Then, every biseparating map T : A → B is a weighted composition operator of the form Tf = h·(f ?φ), where φ is a homeomorphism from Y onto X and h is a nonvanishing element of B. In particular, T is automatically continuous. [ABSTRACT FROM AUTHOR]

We impose a condition on a commutative regular Frechet algebra (A, (pm)) to ensure that A/kerpm is a Frechet Q-algebra. This implies that if θ is an n-homomorphism on certain Frechet algebras (A, (pm)) into semisimple commutative Frechet algebras (B,(qm)) such that θ(kerpm) ⊆ kerqm, for large enough m, then θ is continuous. We also show that if A is a Frechet Q-algebra, B is a semisimple Frechet algebra, θ: A → B is a dense range n-homomorphism such that θ(A) is factorizable, and the spectral radius vB is continuous on the separating space [image omitted](θ), then θ is automatically continuous. [ABSTRACT FROM AUTHOR]

Let A and B be two Banach function algebras and p a two variable polynomial p(z,w)=zw+az+bw+c, (a,b,c∈C). We characterize the general form of a surjection T:A⟶B which satisfies Ranπ(p(Tf,Tg))∩Ranπ(p(f,g))≠∅,(f,g∈A and c≠ab), where Ranπ(f) is the peripheral range of f. [ABSTRACT FROM AUTHOR]