C*-Convexity and the Numerical Range

Magajna, Bojan

Canadian mathematical bulletin, Tome 43 (2000) no. 2, pp. 193-207

Résumé

If $A$ is a prime ${{\text{C}}^{*}}$ -algebra, $a\,\in \,A$ and $\lambda $ is in the numerical range $W\left( a \right)$ of $a$ , then for each $\varepsilon \,>\,0$ there exists an element $h\,\in \,A$ such that $\left\| h \right\|\,=\,1$ and $\left\| {{h}^{*}}(a-\lambda )h \right\|\,<\,\varepsilon $ . If $\lambda $ is an extreme point of $W\left( a \right)$ , the same conclusion holds without the assumption that $A$ is prime. Given any element $a$ in a von Neumann algebra (or in a general ${{\text{C}}^{*}}$ -algebra) $A$ , all normal elements in the weak* closure (the norm closure, respectively) of the ${{\text{C}}^{*}}$ -convex hull of $a$ are characterized.

Détail

DOI : 10.4153/CMB-2000-027-3

Mots-clés : 47A12, 46L05, 46L10