## Banach Journal of Mathematical Analysis

### Analytic aspects of evolution algebras

#### 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 $L_{a}$.

#### Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1543395629

Digital Object Identifier
doi:10.1215/17358787-2018-0018

Mathematical Reviews number (MathSciNet)
MR3895005

Zentralblatt MATH identifier
07002034

#### Citation

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. https://projecteuclid.org/euclid.bjma/1543395629