On invariants of C*-algebras with the ideal property

Abstract

In this paper, we study the relation between the extended Elliott invariant and the Stevens invariant of C*-algebras. We show that in general the Stevens invariant can be derived from the extended Elliott invariant in a functorial manner. We also show that these two invariants are isomorphic for C*-algebras satisfying the ideal property. A C*-algebra is said to have the ideal property if each of its closed two-sided ideals is generated by projections inside the ideal. Both simple, unital C*-algebras and real rank zero C*-algebras have the ideal property. As a consequence, many classes of non-simple C*-algebras can be classified by their extended Elliott invariants, which is a generalization of Elliott’s conjecture.