Weak equivalence to Bernoulli shifts for some algebraic actions

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Namely, we prove that if G is a countable, discrete group and f∈ Mₙ((G)) is invertible on ℓ²(G)⊕ ⁿ, but f is not invertible in Mₙ((G)), then the measure-preserving action of G on Xf equipped with the Haar measure is weakly equivalent to a Bernoulli action. We shall in fact prove this weak equivalence in the case that f has a "formal inverse in ℓ²".