Continuity of homomorphisms into power-associative complete normed algebras

Marcos, Juan Carlos López; Velasco, María V.

doi:10.1515/form.2011.169

Cited by 2 publications

(3 citation statements)

References 6 publications

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“…In [16], I. Kaplansky proved that left division complete normed (non-associative) algebras are isomorphic to C. Many years ago, I. Kaplansky had obtained similar characterizations of the field of complex numbers in [15]. Inspired by these facts, in [22, Definition 2.1] (see also [21]) the following definition of an invertible element was established. This definition is nothing but [3, Proposition 1.19] free of the requirement of associativity.…”

Section: Reviewing the Notion Of Spectrum In The Non-associative Settingmentioning

confidence: 99%

The Jacobson radical of an evolution algebra

Velasco

2018

J. Spectr. Theory

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In this paper we characterize the maximal modular ideals of an evolution algebra A in order to describe its Jacobson radical, Rad(A). We characterize semisimple evolution algebras (i.e. those such that Rad(A) = {0})as well as radical ones. We introduce two elemental notions of spectrum of an element a in an evolution algebra A, namely the spectrum σ A (a) and the m-spectrum σ A m (a) (they coincide for associative algebras, but in general σ A (a) ⊆ σ A m (a), and we show examples where the inclusion is strict). We prove that they are non-empty and describe σ A (a) and σ A m (a) in terms of the eigenvalues of a suitable matrix related with the structure constants matrix of A. We say A is m-semisimple (respectively spectrally semisimple) if zero is the unique ideal contained into the set of a in A such that σ A m (a) = {0} (respectively σ A (a) = {0}). In contrast to the associative case (where the notions of semisimplicity, spectrally semisimplicty and m-semisimplicity are equivalent) we show examples of m-semisimple evolution algebras A that, nevertheless, are radical algebras (i.e. Rad(A) = A). Also some theorems about automatic continuity of homomorphisms will be considered.

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Section: Reviewing the Notion Of Spectrum In The Non-associative Settingmentioning

confidence: 99%

The Jacobson radical of an evolution algebra

Velasco

2018

J. Spectr. Theory

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“…In this section we shall not develop the required framework to consider genetic algebras in a deeper way (a work addressed specifically to this question is [18]). In what follows, we simply provide a flavour of how to apply some of the above results in the framework of evolution algebras.…”

Section: Final Remarks: Potential Application Areas Of This Approachmentioning

confidence: 99%

“…Nevertheless, in Section 5, we give an idea of how to obtain applications of some of these results in the framework of evolution algebras (very relevant algebras in non-Mendelian Genetics, and hence in Molecular Biology). For a wider study addressed specifically to the spectral theory of evolution algebras, see [18].…”

Section: Introductionmentioning

confidence: 99%

The multiplicative spectrum and the uniqueness of the complete norm topology

Marcos

Velasco

2014

Filomat

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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.

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Continuity of homomorphisms into power-associative complete normed algebras

Cited by 2 publications

References 6 publications

The Jacobson radical of an evolution algebra

The Jacobson radical of an evolution algebra

The multiplicative spectrum and the uniqueness of the complete norm topology

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