Amenability and Weak Containment for Actions of Locally Compact Groups on 𝐶*-Algebras

In this work we introduce and study a new notion of amenability for actions of locally compact groups on C ∗ C^* -algebras. Our definition extends the definition of amenability for actions of discrete groups due to Claire Anantharaman-Delaroche. We show that our definition has several characterizations and permanence properties analogous to those known in the discrete case. For example, for actions on commutative C ∗ C^* -algebras, we show that our notion of amenability is equivalent to measurewise amenability. Combined with a recent result of Alex Bearden and Jason Crann, this also settles a long standing open problem about the equivalence of topological amenability and measurewise amenability for a second countable G G -space X X. We use our new notion of amenability to study when the maximal and reduced crossed products agree. One of our main results generalizes a theorem of Matsumura: we show that for an action of an exact locally compact group G G on a locally compact space X X the full and reduced crossed products C 0 (X) ⋊ m a x G C₀ (X) _ maxG and C 0 (X) ⋊ r e d G C₀ (X) _ redG coincide if and only if the action of G G on X X is amenable. We also show that the analogue of this theorem does not hold for actions on noncommutative C ∗ C^* -algebras. Finally, we study amenability as it relates to more detailed structure in the case of C ∗ C^* -algebras that fibre over an appropriate G G -space X X, and the interaction of amenability with various regularity properties such as nuclearity, exactness, and the (L) LP, and the equivariant versions of injectivity and the WEP.