Analytic aspects of evolution algebras

Abstract

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=A1⊕A0, 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 b∈A. We establish necessary and sufficient conditions for convergence of (Lna(b))n, for all b∈A, in terms of the multiplicative spectrum σm(a) of a. Namely, (Lna(b))n converges, for all b∈A, 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.