We first prove that every
σ-unital, purely infinite, simple C∗-algebra is either unital or stable. Consequently,
purely infinite simple C∗-algebras have real rank zero. In particular, the Cuntz
algebras 𝒪n(2 ≤ n ≤ +∞) and the Cuntz-Krieger algebras 𝒪A, where A can be any
irreducible matrix, contain abundant projections. This includes an answer for a
question raised by B. Blackadar in [5, 2.10]. We then prove that the corona and
multiplier algebras associated with many interesting C∗-algebras have real
rank zero. As special cases, we consider the multiplier and corona algebras
associated with certain simple AF algebras, the stabilizations of type II1 and
type III factors, the Cuntz algebras and certain Cuntz-Krieger algebras, the
Bunce-Deddens algebras and some irrational rotation algebras. A recent result
of L. G. Brown and G. K. Pedersen in [12, 3.21] is included as a special
case. In particular, K1(𝒜) = 0, where 𝒜 is a σ-unital, purely infinite simple
C∗-algebra, if and only if the generalized Weyl-von Neumann theorem holds in
M(𝒜⊗𝒦).
