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We investigate approximation properties for $C^*$-algebras and their crossedproducts by actions and coactions by locally compact groups. We show thatHaagerup's approximation constant is preserved for crossed products byarbitrary amenable groups, and we show why this is not always true in thenon-amenable case. We also examine similar questions for other forms of theapproximation property.

APPROXIMATION PROPERTIES FOR CROSSED PRODUCTS BY ACTIONS AND COACTIONS