This paper deals with the automatic continuity of multiplicative polynomial operators on a class of topological algebras. Several results are derived in this direction. We also support our results by some examples.

For two algebras A and B, a linear map T: A® B is called separating, if x.y=0 implies Tx.Ty=0 for all x, yÎ A. The general form and the automatic continuity of separating maps between various Banach algebras have been studied extensively. In this paper, we first extend the notion of separating map for module case and then we give a description of a linear separating map T: B®X, where B is a unital commutative semisimple regular Banach algebra satisfying the Ditkin’s condition and X is a left Banach module over a unital commutative Banach algebra. We also show that if X is hyper semisimple and T is bijective, then T is automatically continuous and T-1 is separating as well.

Let A be a Banach algebra. Let  d, e be linear mappings on A. First we demonstrate a theorem concerning the continuity of double derivations; especially that all of (d, e) -double derivations are continuous on semi-simple Banach algebras, in certain case. Afterwards we define a new vocabulary called “(d, e) - higher double derivation" and present a relation between this subject and derivations and finally give some theorems and corollaries about the automatic continuity of this notion.

We generalize a theorem due to Jarosz, by proving that every almost $n$-multiplicative linear functional on Banach algebra $A$ is automatically continuous. The relation between almost multiplicative and almost $n$-multiplicative linear functional on Banach algebra $A$ is also investigated. Additionally, for commutative Banach algebra $A$, we prove that every almost Jordan homomorphism $\varphi:A\longrightarrow \mathbb{C}$ is an almost $n$-Jordan homomorphism.

In this paper we study higher derivations from JB*-algebras into Banach Jordan algebras. We show that every higher derivation {dm} from a JB*-algebra A into a JB*-algebra B is continuous provided that dO is a *-homomorphism. Also it is proved that every Jordan higher derivation from a commutative C*-algebra or from a C*-algebra which has minimal idempotent and is the closure of its socle is continuous.

In 1989, T. J. Ransford presented a short proof of Johnson's uniqueness of norm theorem. We follow the same method to show that if A and B are Fréchet algebras, B is semisimple and T: A → B is a surjective homomorphism with a certain condition, then T is continuous. In particular, when A is a Banach algebra we conclude that every epimorphism T: A → B is automatically continuous and hence every semisimple Banach algebra has a unique topology as a Fréchet algebra, which is an extension of Johnson's uniqueness of norm theorem.If A and B are Banach algebras, B is semisimple and T: A → B is a dense range homomorphism, then the continuity of T is a long-standing open question. In this work we give a positive answer to this open question with an extra condition on B and then present a partial answer to the well-known Michael's problem. We also obtain similar results for dense range homomorphisms of Fréchet algebras. Finally, we show that if the above question on the continuity of dense range homomorphisms of Banach algebras has a positive answer then the same question has a positive answer for Fréchet algebras.

In this paper, the notation of almost conjugate Jordan homomorphism between Frechet algebras is introduced. It is proven that, under special hypotheses, every almost conjugate Jordan homomorphism on Frechet algebras is an (weakly) almost homomorphism. Also, the automatic continuity of them is generalized.