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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 $A=A_{1}\oplus A_{0}$, 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 $B=\{e_{1},\ldots,e_{n}\}$, then $L_{B}$ is given by $L_{e}$, where $e=\sum_{i}e_{i}$ and $L_{a}$ is the multiplication operator $L_{a}(b)=ab$, for $b\in A$. We establish necessary and sufficient conditions for convergence of $(L_{a}^{n}(b))_{n}$, for all $b\in A$, in terms of the multiplicative spectrum $\sigma_{m}(a)$ of $a$. Namely, $(L_{a}^{n}(b))_{n}$ converges, for all $b\in A$, if and only if $\sigma_{m}(a)\subseteq\Delta\cup\{1\}$ and $\nu(1,a)\leq1$, where $\nu(1,a)$ denotes the index of $1$ in the spectrum of $L_{a}$.

## Citation

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  