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The algebra of all Dirichlet series that are uniformly convergent in the half-plane of complex numbers with positive real part is investigated. When it is endowed with its natural locally convex topology, it is a non-nuclear Fréchet Schwartz space with basis. Moreover, it is a locally multiplicative algebra but not a Q-algebra. Composition operators on this space are also studied.

A map θ:A→B between algebras A and B is called n-multiplicative if θ(a1a2⋯an)=θ(a1) θ(a2)⋯θ(an) for all elements a1,a2,…,an∈A. If θ 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.

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The Fréchet Schwartz Algebra of Uniformly Convergent Dirichlet Series

AUTOMATIC CONTINUITY OF n-HOMOMORPHISMS BETWEEN TOPOLOGICAL ALGEBRAS

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The Fréchet Schwartz Algebra of Uniformly Convergent Dirichlet Series

AUTOMATIC CONTINUITY OF n-HOMOMORPHISMS BETWEEN TOPOLOGICAL ALGEBRAS