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# C$^*$-Convexity and the Numerical Range

Published:2000-06-01
Printed: Jun 2000
• Bojan Magajna
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## Abstract

If $A$ is a prime C$^*$-algebra, $a \in A$ and $\lambda$ is in the numerical range $W(a)$ of $a$, then for each $\varepsilon > 0$ there exists an element $h \in A$ such that $\norm{h} = 1$ and $\norm{h^* (a-\lambda)h} < \varepsilon$. If $\lambda$ is an extreme point of $W(a)$, the same conclusion holds without the assumption that $A$ is prime. Given any element $a$ in a von Neumann algebra (or in a general C$^*$-algebra) $A$, all normal elements in the weak* closure (the norm closure, respectively) of the C$^*$-convex hull of $a$ are characterized.
 MSC Classifications: 47A12 - Numerical range, numerical radius 46L05 - General theory of $C^*$-algebras 46L10 - General theory of von Neumann algebras

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