Journal of Operator Theory

Volume 45, Issue 1, Winter 2001 pp. 39-51.

Limits of vector functionals

Authors: R.J. Archbold (1), and M.H. Shah (2)
Author institution: (1) Department of Mathematical Sciences, University of Aberdeen, King's College, Aberdeen AB24 3UE, Scotland
(2) Department of Mathematical Sciences, University of Aberdeen, King's College, Aberdeen AB24 3UE, Scotland

Summary: For vector functionals on a $C^*$-algebra of operators, we prove an analogue of Glimm's vector state space theorem. We deduce that a\break $C^*$-algebra is prime and antiliminal if and only if the pure functionals are w$^*$-dense in the unit ball of the dual. We also give a necessary and sufficient condition for a convex combination of inequivalent pure functionals to be a w$^*$-limit of pure functionals.

Keywords: $C^{*}$-algebra, vector functional, pure functional, antiliminal, prime

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