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 ⊕ 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 , … , e n } , then L B is given by L e , where e = ∑ i e i and L a is the multiplication operator L a ( b ) = a b , for b ∈ A . We establish necessary and sufficient conditions for convergence of ( L a n ( b ) ) n , for all b ∈ A , in terms of the multiplicative spectrum σ m ( a ) of a . Namely, ( L a n ( b ) ) n converges, for all b ∈ A , if and only if σ m ( a ) ⊆ Δ ∪ { 1 } and ν ( 1 , a ) ≤ 1 , where ν ( 1 , a ) denotes the index of 1 in the spectrum of L a .