Purely infinite corona algebras

Abstract

Let A be a simple, σ−unital, non-unital C*-algebra, with metrizable tracial simplex T(A), projection surjectivity and injectivity, and strict comparison of positive elements by traces. Then the following are equivalent: \nr{i} A has quasicontinuous scale; \nr{ii} M(A) has strict comparison of positive elements by traces; \nr{iii} M(A)/A is purely infinite; \nr{iii'} M(A)/Imin is purely infinite; \nr{iv} M(A) has finitely many ideals; \nr{v} Imin=Ifin. If furthermore Mn(A) has projection surjectivity and injectivity for every n, then the above conditions are equivalent to: \nr{vi} V(M(A)) has finitely many order ideals.