C*-ALGEBRAS WITH REAL RANK ZERO AND THEIR CORONA AND MULTIPLIER ALGEBRAS PART IV

Abstract

By proving various equivalent versions of the generalized Weyl-von Neumann theorem, we investigate the structure of projections in the multiplier algebra [Formula: see text] of certain C*-algebra [Formula: see text] with real rank zero. For example, we prove that [Formula: see text] if and only if any two projections in [Formula: see text] are simultaneously quasidiagonal. In case [Formula: see text] is a purely infinite simple C*-algebra, [Formula: see text] if and only if any two projections in [Formula: see text] are simultaneously quasidiagonal. If [Formula: see text] is one of the Cuntz algebras, or one of finite factors or type III factors, then any two projections in [Formula: see text] are simultaneously quasidiagonal. On the other hand, if [Formula: see text] is one of the Bunce-Deddens algebras or one of the irrational rotation algebras of real rank zero, then there exist two projections in [Formula: see text] which are not simultaneously quasidiagonal.