Analytic aspects of evolution algebras

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Title: Analytic aspects of evolution algebras
Authors: Mellon, PaulineVelasco, María Victoria
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Date: 28-Nov-2019
Online since: 2019-12-10T14:13:52Z
Abstract: We prove that every evolution algebra A is a normed algebra, for an l1-norm defined in terms of a fixed natural basis. We further show that a normed evolution algebra A is a Banach algebra if and only if A=A1⊕A0, where A1 is finite-dimensional and A0 is a zero-product algebra. In particular, every nondegenerate Banach evolution algebra must be finite-dimensional and the completion of a normed evolution algebra is therefore not, in general, an evolution algebra. We establish a sufficient condition for continuity of the evolution operator LB of A with respect to a natural basis B, and we show that LB need not be continuous. Moreover, if A is finite-dimensional and B={e1,…,en}, then LB is given by Le, where e=∑iei and La is the multiplication operator La(b)=ab, for b∈A. We establish necessary and sufficient conditions for convergence of (Lna(b))n, for all b∈A, in terms of the multiplicative spectrum σm(a) of a. Namely, (Lna(b))n converges, for all b∈A, if and only if σm(a)⊆Δ∪{1} and ν(1,a)≤1, where ν(1,a) denotes the index of 1 in the spectrum of La.
Funding Details: European Commission - European Regional Development Fund
metadata.dc.description.othersponsorship: University College Dublin School of Mathematics and Statistics
Spanish Ministry of Economy, Industry and Competitiveness
Junta de Andalucía
Type of material: Journal Article
Publisher: Duke University Press
Journal: Banach Journal of Mathematical Analysis
Volume: 13
Issue: 1
Start page: 113
End page: 132
Copyright (published version): 2019 Tusi Mathematical Research Group
Keywords: Evolution algebraEvolution operatorGenetic algebra
DOI: 10.1215/17358787-2018-0018
Language: en
Status of Item: Peer reviewed
Appears in Collections:Mathematics and Statistics Research Collection

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