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.

Keywords: Pure state, pure functional, irreducible representation, $C^*$-algebra,\break spectrum

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