Crossed products by spectrally free actions

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 α of a finite group G  , spectral freeness is equivalent to strong pointwise outerness, and also to the condition that Γ˜(αg)≠{1} for every g∈G∖{1}g∈G∖{1}.We then prove permanence results for reduced crossed products by exact spectrally free actions, for crossed products by arbitrary actions of Z/2ZZ/2Z, 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.