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.

$K\sb 1$-groups, quasidiagonality, and interpolation by multiplier projections

“…AF algebras, the Calkin algebra, von Neumann algebras, and Bunce-Deddens algebras [5; 8, §4] have the FS property. Various other examples of such C* -algebras will be given in subsequent papers of the author [30]. It has recently been proved [13] that A has FS if and only if A®K has FS; again iff A has real rank zero.…”

“…Thus, the stabilized Bunce-Deddens algebras and the Calkin algebra provide basic counterexamples to the conjecture of G. K. Pedersen " A has FS => M(A) has FS". Various nontrivial counterexamples for " M(A)/A has FS but M (A) does not" will be given in subsequent papers [30].…”

Real rank and topological dimension of higher rank graph algebras

“…So we restrict our attention to the special case of k-graphs Λ for which C * (Λ) is purely infinite. Simple purely infinite C * -algebras are automatically of real-rank zero by [4,40], but nonsimple purely infinite C * -algebras need not have real-rank zero (see Examples 6.1 and 6.2). Indeed, Pasnicu and Rørdam have proved that a purely infinite C * -algebra has realrank zero if and only if it has topological dimension zero and satisfies a K-theoretic criterion called K 0 -liftability [31].…”

Structure of Nonunital Purely Infinite Simple Rings

“…A large series of contributions, due to Blackadar, Brown, Lin, Pedersen, Phillips, Rørdam and Zhang, among others, reflect the interest in the structure of such algebras. A particular interest deserves Zhang's result [8], dividing σ-unital purely infinite simple C*-algebras in two types: unital and stable. This result, known as Zhang's Dichotomy for σ-unital purely infinite simple C*-algebras, played a central role in the study of the structure of corona and multiplier algebras for C*-algebras with real rank zero.…”