On the norm and spectral radius of Hermitian elements (Q946804)

An element \(a\) of a complex unital Banach algebra \({\mathcal A}\) is called \textit{Hermitian} if \(\| \exp(ita)\| =1\) for all \(t \in \mathbb R\) or, equivalently, if the numerical range of \(a\) is real. It is known that if \(a \in\mathcal{A}\) is Hermitian, then \(\| a^n\| = |a^n|\) for every \(n\in {\mathbb N}\), where \(|x|\) denotes the spectral radius of \(x \in\mathcal{A}\). A function \(F: \mathbb R \longrightarrow \mathbb C\) is called a \textit{universal symbol} if \(\| F(a)\| = | F(a)|\) for every \(\mathcal{A}\) and all Hermitian \(a \in \mathcal{A}\). In this paper, the author characterizes universal symbols in terms of positive-definite functions.