Journal of Operator Theory

Volume 63, Issue 2, Spring 2010 pp. 349-362.

Limits of pure functionals of

$C^*$ -algebras

Authors: M.H. Shah
Author institution: Department of Mathematics, Lahore University of Management Sciences (LUMS), 54792-Lahore, Pakistan

Summary: It is shown that, for a

$C^*$ -algebra

$A,$ every (weak*-) limit of pure functionals is a multiple of a pure functional if and only if every limit of pure states is a multiple of pure states (a condition previously studied by Glimm). On the other hand, it is shown that the set of pure states

$P(A)$ being closed does not force the set of pure functionals

$G(A)$ to be closed. The conditions

$\overline{G(A)}$ =

$G(A)$ and

$\overline{G(A)} = G(A)\cup \{0\}$ are characterised in terms of sums of homogeneous

$C^*$ -algebras.

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