Braatvedt, G.; Brits, R.; Raubenheimer, H. Spectral characterizations of scalars in a Banach algebra. (English) Zbl 1193.46028 Bull. Lond. Math. Soc. 41, No. 6, 1095-1104 (2009). For a complex Banach algebra \(A\) and \(a \in A\), let \(\sigma (a)\) denote the spectrum of \(a\), and as usual, if \(T\) is a set, \( \# T\) stands for the cardinality of \(T\). The main result in the present paper says that if \(A\) is a unital semisimple Banach algebra and \(a \in A\) is such that \( \# \sigma (ax) \leq \# \sigma (x)\) for all \(x\) in some neighborhood of \(1\), then \(a\) is a scalar. This extends a result of R. Brits [Quaest. Math 31, 179–188 (2008; Zbl 1168.46027)]. The authors apply their result to show that a unital semisimple Banach algebra \(A\) is commutative if \(\# \sigma (xyz)= \# \sigma(yzx)\) for all \(x,y,z\) in some neighborhood of \(1\). Moreover, they study elements \(a\) in a Banach algebra \(A\) satisfying \(r(ax)= r(x)\) for all \(x \in A\), or \( \| ax \|= \| x \|\) for all \(x \in A\); here, \(r(x)\) stands for the spectral radius of \(x\). Reviewer: Nadia Boudi (Meknes) Cited in 6 Documents MSC: 46H05 General theory of topological algebras 31C05 Harmonic, subharmonic, superharmonic functions on other spaces Keywords:Banach algebras; spectrum; spectral radius; centre; commutativity Citations:Zbl 1168.46027 PDFBibTeX XMLCite \textit{G. Braatvedt} et al., Bull. Lond. Math. Soc. 41, No. 6, 1095--1104 (2009; Zbl 1193.46028) Full Text: DOI