Boundaries of weak peak points in noncommutative algebras of Lipschitz functions
Let \(\mathbb{F}\) be one of the fields \(\mathbb{R}\) of real numbers, \(\mathbb{C}\) of complex numbers or the non-commutative division ring \(\mathbb{H}\) of quaternions, \(X\) a compact Hausdorff space, \(C(X,\mathbb{F})\) the space of all continuous \(\mathbb{F}\)-valued functions on \(X\), \(M(f)=\{x\in X:|f(x)|=\|f\|_\infty\}\), \(\mathcal{A}\subset C(X,\mathbb{F})\) a subalgebra, \(E\subset X\) an \(m\)-set (that is a nonempty set such that \(E=\bigcap_{f\in S}M(f)\)) for some family of functions \(S\subset \mathcal{A}\)), \(\varepsilon^\circ \) the collection of all minimal \(m\)-sets of \(\mathcal{A}\) and \(\delta\mathcal{A}=\bigcup _{E\in \varepsilon^\circ}E\). If all the minimal \(m\)-sets of \(\mathcal{A}\) are singletons, then \(\delta\mathcal{A}\) is a boundary for \(\mathcal{A}\) consisting exactly of weak peak points. It is shown that, if every \(m\)-set of \(\mathcal{A}\subset C(X,\mathbb{F})\) is a singleton, then \(\delta\mathcal{A}\) is contained in every closed boundary of \(\mathcal{A}\) and the intersection of all closed boundaries is a closed boundary. Moreover, sufficient conditions for a point-separating algebra \(\mathcal{A}\subset C(X,\mathbb{F})\) are given that a minimal \(m\)-set is a singleton and a characterization of a class of algebras over \(\mathbb{R}\) of \(\mathbb{F}\)-valued Lipschitz functions is given for which there exists a minimal closed boundary. In addition, a new proof for the statement that an associative, unital normed algebra \(\mathcal{A}\) over \(\mathbb{R}\) is topologically isomorphic to \(\mathbb{R}\) or \(\mathbb{C}\) or \(\mathbb{H}\) if \(\|fg\|=\|f\| \|g\|\) for all \(f,g\in \mathcal{A}\).
scientific article; zbMATH DE number 6039962
scientific article