Relating properties of crossed products to those of fixed point algebras

For a number of properties of C*-algebras, including real rank zero, stable rank one, pure infiniteness, residual hereditary infiniteness, the combination of pure infiniteness and the ideal property, the property of being an AT algebra with real rank zero, and D-stability for a separable unital strongly selfabsorbing C*-algebra D, we prove the following. Let A be a separable C*-algebra, let G be a second countable compact abelian group, and let α:G→Aut(A) be any action of G on A. Then the fixed point algebra A^α has the given property if and only if the crossed product C*(G, A, α) has the same property.