  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 stability under tensoring with a strongly
selfabsorbing C*-algebra, we prove the following. Consider an arbitrary action
of a second countable compact abelian group on a separable C*-algebra. Then the
fixed point algebra under the action has the given property if and only if the
crossed product has the same property.
