AUTOMATIC CONTINUITY OF n-HOMOMORPHISMS BETWEEN TOPOLOGICAL ALGEBRAS

A map theta : A -> B between algebras A and B is called n-multiplicative if theta(a(1)a(2)...a(n)) = theta(a(1)) theta(a(2))...theta(a(n)) for all elements a(1), a(2,)..., a(n) epsilon A. If theta is also linear then it is called an n-homomorphism. This notion is an extension of a homomorphism. We obtain some results on automatic continuity of n-homomorphisms between certain topological algebras, as well as Banach algebras. The main results are extensions of Johnson's theorem to surjective n-homomorphisms on topological algebras, a theorem due to C. E. Rickart in 1950 to dense range n-homomorphisms on topological algebras and two theorems due to E. Park and J. Trout in 2009 to *-preserving n-homomorphisms on lmc *-algebras.