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Spectral characterizations of scalars in a Banach algebra. (English) Zbl 1193.46028

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 \(x\).

MSC:

46H05 General theory of topological algebras
31C05 Harmonic, subharmonic, superharmonic functions on other spaces

Citations:

Zbl 1168.46027
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