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 f\"ur Mathematik, Humboldt Universit\"at 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

$X$ .

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