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 La and Ra 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.
Automatic Continuity of Dense Range Homomorphisms into Multiplicatively Semisimple Complete Normed Algebras
