The multiplicative spectrum and the uniqueness of the complete norm topology Marcos Sánchez, Juan Carlos Velasco Collado, María Victoria Spectrum Spectral radius Non-associative complete normed algebra Radical Homomorphism Continuity Genetic algebra 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. 2014-11-06T08:55:37Z 2014-11-06T08:55:37Z 2014 info:eu-repo/semantics/article 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] 0354-5180 http://hdl.handle.net/10481/33543 10.2298/FIL1403473M eng http://creativecommons.org/licenses/by-nc-nd/3.0/ info:eu-repo/semantics/openAccess Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License Univerzitet U Nišu