The multiplicative spectrum and the uniqueness of the complete norm topology
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Univerzitet U Nišu
SpectrumSpectral radiusNon-associative complete normed algebraRadicalHomomorphismContinuityGenetic algebra
Marcos Sánchez, J.C.; Velasco Collado, M.V. The multiplicative spectrum and the uniqueness of the complete norm topology. Filomat, 28(3): 473-485 (2014). [http://hdl.handle.net/10481/33543]
SponsorshipResearch supported by Junta de Andaluc´ıa grant FQM 0199.
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.