Purely infinite corona algebras

Abstract

Let A be a simple, sigma-unital, non-unital C*-algebra, with metrizable tracial simplex T(A), which is projection-surjective and injective and has strict comparison of positive elements by traces. Then the following are equivalent: (i) A has quasicontinuous scale; (ii) The multiplier algebra M(A) has strict comparison of positive elements by traces; (iii) The coronal algebra M(A)/A is purely infinite; (iii') The quotient M(A)/Imin is purely infinite; (iv) M(A) has finitely many ideals; (v) Imin=Ifin. If furthermore algebra of n by n matrices of A is projection-surjective and injective for every n, then the above conditions are equivalent to: (vi) the monoid V(M(A)) has finitely many order ideals. Quasicontinuity of the scale is a notion introduced by Kucerovsky and Perera that extends both the property of having finitely many extremal traces and of having continuous scale. Projection-surjectivity and injectivity permit to identify projections in M(A) that are not in A with lower semicontinuous affine functions on T(A). Imin is the smallest ideal of M(A) properly containing A, and Ifin is the ideal of of M(A) generated by the positive elements with evaluation functions finite over the extremal boundary of T(A).