Some extremal algebras for Hermitians (Q5890198)

Let \(A\) be a complex unital Banach algebra with dual space \(A'\). For \(a\in A\) the compact set \(V(a):=\{ \varphi (a): \varphi\in A', \|\varphi\|=\varphi(1)=1\}\) is called the numerical range of \(a\). An element \(a\) is said to be Hermitian if \(V(a)\subset {\mathbf R}\), equivalenty \(\|\exp(ita)\|=1\), for all \(t\in{\mathbf R}\). For any compact convex set \(K\subset{\mathbf C}\) there is a unital Banach algebra \(Ea(K)\) generated by an element \(h\) in which every polynomial in \(h\) attains its maximum norm over all Banach algebras subject to its numerical range \(V(h)\) being contain in \(K\) [Acta Math. 128, 123-142 (1972; Zbl 0229.46049)]. Such algebra \(Ea(K)\) is called the extremal algebra for \(K\). NEWLINENEWLINENEWLINEIn this paper the authors study three extremal Banach algebras \(Ea(K)\): NEWLINENEWLINENEWLINE(a) generated on two non-commuting generators \(u\), \(v\) with \(u^2=v^2=1\) and \(u\), \(v\) Hermitian; NEWLINENEWLINENEWLINE(b) generated by an element of norm \(1\) all of whose odd positive powers are Hermitian; NEWLINENEWLINENEWLINE(c) generated an element (not Hermitian) of norm \(1\) all of whose even positive powers are Hermitian. NEWLINENEWLINENEWLINEIn all three cases isometrically isomorphic dublicates of these extremal algebras are indicated. The third algebra is identified with a space of functions. In addition, the authors calculate the norm and numerical range for various elements.