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Let $M$ be a compact manifold and let $D$ be a minimal diffeomorphism from $M$ to itself (meaning there are no nontrivial invariant subspaces). I believe it was Connes who proved that if the first cohomology group of $M$ vanishes then the crossed product C* algebra $C(M) \times_D \mathbb{Z}$ has no nontrivial idempotents.

I am wondering if this theorem has a (partial) converse. Ideally I would like a procedure which, under suitable assumptions on $M$ and $D$, associates a nontrivial projection to each cohomology class of $M$. I'm hoping this procedure would give an alternative way to construct the Powers-Rieffel projections associated to irrational rotation algebras, for example.

Any ideas?

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