Journal of Operator Theory

Volume 55, Issue 2, Spring 2006 pp. 239-251.

The range of generalized Gelfand transforms on $C^*$-algebras

Authors: Eberhard Kirchberg
Author institution: Institut für Mathematik, Humboldt Universität zu Berlin, Unter den Linden 6, D-10099 Berlin, Germany

Summary: It is shown that every Dini function on the primitive ideal space of a $C^*$-algebra $A$ is the generalized Gelfand transform of an element of $A$. Here a Dini function on a topological space $X$ means a non-negative lower semi-continuous function $f$ on $X$ with $\sup f\Big(\bigcap\limits _\tau F_\tau\Big)=\inf\limits_\tau\sup f(F_\tau)$ for every downward directed net $\{F_\tau\}_\tau$ of closed subsets of $X$.

Keywords: $C^*$-algebras, generalized Gelfand transforms, Dini functions, general topology, Hausdorff lattices.

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