PURELY INFINITE CORONA ALGEBRAS

Let A be a simple, sigma-unital, non-unital Of C*-algebra, with metriz- able tracial simplex T(A), projection surjectivity and injectivity, and strict comparison of positive elements by traces. Then the following are equivalent: (i) A has quasicontinuous scale; (ii) M (A) has strict comparison of positive elements by traces; (iii) M (A) / A is purely infinite; (iii') M (A) / I-min is purely infinite; (iv) M (A) has finitely many ideals; (v) I-min =I- (fin.) If furthermore M-n (A) has projection surjectivity and injectivity for every n, then the above conditions are equivalent to: (vi) V (M (A)) has finitely many order ideals.