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Analytic aspects of evolution algebras

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Download Evolution Algebras-Mellon-Velasco-Archive.pdf380.12 KB
Author(s)
Mellon, Pauline 
Velasco, María Victoria 
Uri
http://hdl.handle.net/10197/11243
Date Issued
28 November 2019
Date Available
10T14:13:52Z December 2019
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.
Sponsorship
European Commission - European Regional Development Fund
Other Sponsorship
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 algebra

  • Evolution operator

  • Genetic algebra

DOI
10.1215/17358787-2018-0018
Language
English
Status of Item
Peer reviewed
ISSN
1735-8787
This item is made available under a Creative Commons License
https://creativecommons.org/licenses/by-nc-nd/3.0/ie/
Owning collection
Mathematics and Statistics Research Collection
Scopus© citations
5
Acquisition Date
Feb 6, 2023
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Views
660
Acquisition Date
Feb 7, 2023
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Downloads
176
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Acquisition Date
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