The multiplicative spectrum and the uniqueness of the complete norm topology

Abstract

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.

The Jacobson radical of an evolution algebra

“…In [22] (see Theorem 3.5 and Corollary 3.6) we proved the following result in the general non-associative setting. Particularly, whenever A is associative, we obtain as a corollary the well known theorem of B. E. Johnson [14] (see also [3,10,26]) that in words of T. Palmer is a "cornerstone of the Banach algebra theory".…”

“…The notion of m-semisimplicity was used in [22] to prove the automatic continuity of every surjective homomorphism from a Banach algebra onto a m-semisimple Banach algebra.…”

Pulse Processes in Networks and Evolution Algebras

“…As proved in [29], Proposition 2.2, for an arbitrary complex algebra A and a ∈ A, we have that, if A has not a unit then…”

“…On the other hand, as proved in [29], Proposition 2.5, if (A, • ) is a non-associative Banach algebra then σ A m (a) is a set of complex numbers such that |λ| ≤ a , for every a ∈ A. In fact, the m-spectrum extends the classical notion of spectrum of an element in an associative algebra to the non-associative framework by keeping a good number of its essential properties.…”

“…In [29], the following notion of spectrum of an element in a non-necessarily associative algebra was introduced, and therefore many results of the spectral theory of Banach algebras were extended to the non-associative framework. Definition 5.…”