The Multiplicative Spectrum and the Uniqueness of the Complete Norm Topology

J. C. Marcos, M. V. Velasco

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.