Normal spectral approximation in \(C^*\)-algebras and in von Neumann-algebras. (English) Zbl 0851.46039
Summary: The general problem of normal spectral approximation in a unital \(C^*\)-algebra \({\mathcal A}\) is studied. If \(\Delta\) is a closed and convex subset of the complex plane then the distance from a normal element \(a\) of \({\mathcal A}\) to the set \({\mathcal N}_{\mathcal A} (\Delta)\) of all normal elements with spectrum in \(\Delta\) equals the one-sided Hausdorff distance from \(\sigma (a)\) to \(\Delta\). As a consequence every normal element \(a\) has an \({\mathcal N}_{\mathcal A} (\Delta)\)-approximant which is a function of \(a\). Furthermore the approximant is unique whenever every point of \(\sigma (a)\) has the same distance to \(\Delta\). These results are shown for the approximation in the \(C^*\)-norm as well as in another topological equivalent norm.
Keywords:
normal spectral approximation; unital \(C^*\)-algebra; one-sided Hausdorff distance; \(C^*\)-normReferences:
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