Commutators with finite spectrum
Let \(d\) be a continuous derivation on a complex Banach algebra \(A\). In this paper, the authors prove that the following conditions are equivalent (here \(\# S\) denotes the cardinality of a set \(S\) and \(\sigma(T)\) denotes the spectrum of an operator \(T\)): (a) \(\#\sigma(dx)<\infty\) for all \(x\in A\); (b) \(\#\sigma([x,dx])<\infty\) for all \(x\in A\), where \([x,dx]=x(dx)-(dx)x\); (c) there exists \(a\in A\) such that \(a+{\text{rad}\,} A\in {\text{soc}\,}(A/{\text{rad}\,} A)\) for all \(x\in A\); note that for a semisimple algebra \(A\), the socle \({\text{soc}\,}A\) is defined as the sum of all minimal left ideals of \(A\).
scientific article; zbMATH DE number 2138882