Pure functionals and irreducible representations of C*-algebras
Basic properties of pure functionals of a C*-algebra are reviewed, and this is followed by an investigation of equivalent representations of a pure functional, restriction to ideals, and extension to bigger C*-algebras. The relationship between notions of regularity for points in the spectrum of a C*-algebra is studied. A localised version of Fell-Dixmier conditions for continuous trace of a C*-algebra is obtained. The weak*-closure of the space of pure functionals of arbitrary and homogeneous C*-algebras is investigated. An analogue of Glimm's Vector State Space Theorem is proved. It is shown that G(A) = A*1 if and only if A is prime and antiliminal. Some results about the limits of pure functionals of an arbitrary C*-algebra are obtained.