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January 2019 Analytic aspects of evolution algebras
P. Mellon, M. Victoria Velasco
Banach J. Math. Anal. 13(1): 113-132 (January 2019). DOI: 10.1215/17358787-2018-0018

Abstract

We prove that every evolution algebra A is a normed algebra, for an l 1 -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 = A 1 A 0 , where A 1 is finite-dimensional and A 0 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 L B of A with respect to a natural basis B , and we show that L B need not be continuous. Moreover, if A is finite-dimensional and B = { e 1 , , e n } , then L B is given by L e , where e = i e i and L a is the multiplication operator L a ( b ) = a b , for b A . We establish necessary and sufficient conditions for convergence of ( L a n ( b ) ) n , for all b A , in terms of the multiplicative spectrum σ m ( a ) of a . Namely, ( L a n ( b ) ) n converges, for all b A , if and only if σ m ( a ) Δ { 1 } and ν ( 1 , a ) 1 , where ν ( 1 , a ) denotes the index of 1 in the spectrum of L a .

Citation

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P. Mellon. M. Victoria Velasco. "Analytic aspects of evolution algebras." Banach J. Math. Anal. 13 (1) 113 - 132, January 2019. https://doi.org/10.1215/17358787-2018-0018

Information

Received: 10 February 2018; Accepted: 26 May 2018; Published: January 2019
First available in Project Euclid: 28 November 2018

zbMATH: 07002034
MathSciNet: MR3895005
Digital Object Identifier: 10.1215/17358787-2018-0018

Subjects:
Primary: 58C40
Secondary: 34L05, 35P05

Rights: Copyright © 2019 Tusi Mathematical Research Group

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Vol.13 • No. 1 • January 2019
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