Crossed products by spectrally free actions

Abstract

We define spectral freeness for actions of discrete groups on C*-algebras. We relate spectral freeness to other freeness conditions; an example result is that for an action of a finite group, spectral freeness is equivalent to strong pointwise outerness, and also to the condition that the strong Connes spectrum of the action of the integers generated by a nontrivial group element is always nontrivial. We then prove permanence results for reduced crossed products by exact spectrally free actions, for crossed products by arbitrary actions of the two element group, and for extensions, direct limits, stable isomorphism, and several related constructions, for the following properties: The combination of pure infiniteness and the ideal property. Residual hereditary infiniteness (closely related to pure infiniteness). Residual (SP) (a strengthening of Property (SP) suitable for nonsimple C*-algebras). The weak ideal property (closely related to the ideal property). For the weak ideal property, we can allow arbitrary crossed products by any finite abelian group. These properties of C*-algebras are shown to have formulations of the same general type, allowing them all to be handled using a common set of theorems.