Analytic aspects of evolution algebras
Finite-dimensional evolution algebras were introduced by \textit{J. P. Tian} and \textit{P. Vojtěchovský} [Quasigroups Relat. Syst. 14, No. 1, 111--122 (2006; Zbl 1112.17001)]. \par In the present paper, the authors deal with evolution algebras of arbitrary dimension, as they were studied by \textit{Y. Cabrera Casado} et al. [Linear Algebra Appl. 495, 122--162 (2016; Zbl 1395.17079)]. An evolution algebra is an algebra $A$ equipped with a basis $B=\{e_i: i \in \Lambda\}$ such that $e_ie_j=0$ for all $i \neq j$ (called a natural basis of $A$). The authors study normed (in particular, Banach) evolution algebras. They give a necessary and sufficient condition so that an evolution algebra is a normed one, endowed with the $l_1$-norm (with respect to a natural basis). They also prove that a normed evolution algebra $A$ is Banach if it splits into a direct sum of a finite-dimensional evolution algebra and a (possibly infinite-dimensional) zero-product subalgebra of $A$. They also assert, without proof, that the converse is also true. By an example, they show that the completion of a normed evolution algebra is not, in general, itself an evolution algebra (for the same underlying product). Related to this, and using the concept of a nondegenerate evolution algebra (here, the elements of some natural basis satisfy $e_i^2 \neq 0$), the authors prove that nondegenerate Banach evolution algebras are finite-dimensional. The last statement implies that the completion of a nondegenerate infinite-dimensional normed evolution algebra is not an evolution one. \par Furthermore, the authors study the continuity of the evolution operator associated to a natural basis of an evolution algebra $A$, namely, of the unique linear map $L_B : A \rightarrow A$ with $L_B (e_i)=e_i^2$. They give an example of a normed evolution algebra such that $L_B$ is not continuous, and then they supply a sufficient condition for the continuity of $L_B$. \par In the finite-dimensional case, they employ certain elements of normed evolution algebras, called equilibrium generators. If the $m$-spectral radius of an element dominates $1$, then that element is not an equilibrium generator. Among the main results, they characterize an equilibrium generator $a$ via the multiplicative (point) spectrum and the index of 1 as an eigenvalue of the multiplication operator $L_a$.
scientific article; zbMATH DE number 7002034
scientific article