Crossed products, the weak ideal property, and topological dimension zero

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Let α be an action of a finite group G on a C*-algebra A and assume that A has a composition series consisting of α-invariant ideals in which every quotient of successive ideals is α-simple (this happens, e.g., if A has finitely many α-invariant ideals). We prove that in this case, the crossed product C⁎(G,A,α) has the weak ideal property ⇔ the fixed point algebra Aα has the weak ideal property ⇔ A has the weak ideal property. We show that crossed products of many C*-algebras with the ideal property by finite groups have the weak ideal property. We prove that if G is a second countable locally compact group with a closed normal subgroup H such that the group G/H is finite and abelian, then a crossed product of a separable C*-algebra by G has the weak ideal property (respectively, topological dimension zero) ⇔ the crossed product given by the restriction of the action to H has the weak ideal property (respectively, topological dimension zero). We show that, many times, if the reduced group C*-algebra of a locally compact group G has the weak ideal property, or if it has topological dimension zero, then the same is true of the reduced group C*-algebra of many subquotients of G.