On the norm and numerical radius of Hermitian elements (Q1901199)

Let \(A\) be a complex Banach algebra with unit 1. For \(a\in A\), it is known that: \[ |a|\leq\nu(a)\leq|a|, \] where \(|a|=\) spectral radius, \(\nu(a)\) the numerical radius of \(a\). Let \(a\in A\) be a Hermitian element such that the spectrum of \(a\) is \(\text{Sp}(a)= [-\sigma,\sigma]\), \(\sigma>0\). If \(\langle a\rangle\) is the algebra generated by \(a\), then the maximal norm on \(\langle a\rangle\), generates an extremal algebra denoted by \(E_a[-\sigma,\sigma]\). The dual \((E_a[-\sigma,\sigma])^*\) of the algebra \(E_a[-\sigma,\sigma]\) can be identified with a certain subspace of \(L_\infty({\mathbb{R}})\), which is determined as the set \(B_\sigma\) of all entire functions \(f\) satisfying the condition \[ |f(z)|\leq M e^{\sigma|\text{Im }z|}, \] for all complex \(z\), and some \(M\geq 0\). For \(f\in B_\sigma\), denote \(N_f= \{x\in{\mathbb{R}}:|f(x)|=|f|\}\), a function \(f\in B_\sigma\), is called real if \(f(x)\in{\mathbb{R}}\), \(\forall x\in{\mathbb{R}}\). An element \(b\in E_a[-\sigma,\sigma]\) is called a real element if it possesses at least one corresponding real measure. Lemma: For any real \(f\in B_\sigma\), there exists a real function \(g\in B_\sigma\), such that \(g(x)=0\), \(\forall x\in N_f\) and \[ |f|= |f+ig|=|(f+ ig)(0)|. \] The author proved the above lemma and used it to prove the following Theorem: The numerical radius of any real element of the algebra \(E_a[-\sigma,\sigma]\) is equal to its norm, i.e., \(\nu(a)=|a|\).