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Journal of Operator Theory

Volume 52, Issue 2, Fall 2004  pp. 385-420.

Global Glimm halving for $C^*$-bundles

Authors Etienne Blanchard (1) and Eberhard Kirchberg (2)
Author institution: (1) Institut de Mathematiques, Projet Algebres d'operateurs (Plateau 7E), 175, rue du Chevaleret, F-75013 Paris, France
(2) Institut fuer Mathematik, Humboldt Universitaet zu Berlin, Unter den Linden 6, D--10099 Berlin, Germany

Summary:  A global notion of Glimm halving for $C^*$-algebras is considered which implies that every nonzero quotient of an algebra with this property is antiliminal. We prove subtriviality and selection results for Banach spaces of sections vanishing at infinity of a continuous field of Banach spaces. We use them to prove the global Glimm halving property for strictly antiliminal $C^*$-algebras with Hausdorff primitive ideal space of finite dimension. This implies that a $C^*$-algebra $A$ with Hausdorff primitive ideal space of finite dimension must be purely infinite if its simple quotients are purely infinite.

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