Banach Journal of Mathematical Analysis

Analytic aspects of evolution algebras

P. Mellon and M. Victoria Velasco


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=A1A0, 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 bA. We establish necessary and sufficient conditions for convergence of (Lan(b))n, for all bA, in terms of the multiplicative spectrum σm(a) of a. Namely, (Lan(b))n converges, for all bA, 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.

Article information

Banach J. Math. Anal., Volume 13, Number 1 (2019), 113-132.

Received: 10 February 2018
Accepted: 26 May 2018
First available in Project Euclid: 28 November 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58C40: Spectral theory; eigenvalue problems [See also 47J10, 58E07]
Secondary: 34L05: General spectral theory 35P05: General topics in linear spectral theory

evolution algebra evolution operator genetic algebra


Mellon, P.; Velasco, M. Victoria. Analytic aspects of evolution algebras. Banach J. Math. Anal. 13 (2019), no. 1, 113--132. doi:10.1215/17358787-2018-0018.

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