In 1967, Barry E. Johnson proved the uniqueness of the complete norm topology for semisimple Banach algebras as well as the automatic continuity of homomorphisms from a Banach algebra onto a semisimple Banach algebra. In this paper, we show that the associativity of the product is superfluous in these results.

Abstract.We define the spectrum of an element a in a non-associative algebra A according to a classical notion of invertibility (a is invertible if the multiplication operators L a and R a are bijective). Around this notion of spectrum, we develop a basic theoretical support for a non-associative spectral theory. Thus we prove some classical theorems of automatic continuity free of the requirement of associativity. In particular, we show the uniqueness of the complete norm topology of m-semisimple algebras, obtaining as a corollary of this result a well-known theorem of Barry E. Johnson (1967).The celebrated result of C.E. Rickart (1960) about the continuity of dense-range homomorphisms is also studied in the non-associative framework.Finally, because non-associative algebras are very suitable models in genetics, we provide here a hint of how to apply this approach in that context, by showing that every homomorphism from a complete normed algebra onto a particular type of evolution algebra is automatically continuous.

Let A and B be complete normed algebras with a unit such that B is simple and power-associative, and let  W A ! B be a dense-range homomorphism. We prove that if  is irreducible (that is, Â.I / ¤ B, for every closed proper ideal I of A), then  is continuous. The continuity of non-irreducible homomorphisms is also obtained provided that the set of «spectrally rare»elements in the range algebra is not dense in B. These results extend the classical Rickart's dense-range homomorphism theorem to the non-associative setting.

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