Is it possible to find a non-amenable group G, an elementary C*-algebra A (i.e A=K(H) for some Hilbert space) with an action of G on A, such that the corresponding semi-direct bundle (or called C*-dynamical system) is amenable (i.e the homomorphism from the full crossed product to the reduced crossed product is injective)?
The following idea is my attempt: let G be a non-amenable group, and let G act on itself by left multiplication, i. e t \mapsto s^{-1}t for each s in G. This action is amenable, and we can use this action to define an action of G on K(L_2(G)) by natural way. Is the semi-direct bundle consisting of this G, K(L_2(G)) and the action amenable?
Have a look at https://arxiv.org/pdf/math/9801069.pdf
https://math.stackexchange.com/questions/1494533/are-semi-direct-products-and-fiber-products-similar-constructions-in-different-c
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