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 $B=\{e_1,\dots ,e_n\}$, then $L_B$ is given by $L_e$, where $e={\sum }_i{e}_i$ and $L_a$ is the multiplication operator $L_a\left(b\right)=ab$, for $b\in A$. We establish necessary and sufficient conditions for convergence of $\left(L_a^n\right(b){)}_n$, for all $b\in A$, in terms of the multiplicative spectrum ${\sigma }_m\left(a\right)$ of $a$. Namely, $\left(L_a^n\right(b){)}_n$ converges, for all $b\in A$, if and only if ${\sigma }_m\left(a\right)\subseteq \Delta \cup \left\{1\right\}$ and $\nu (1,a)\le 1$, where $\nu (1,a)$ denotes the index of $1$ in the spectrum of $L_a$.