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Geometrically, we may think of Sq2n−1 as a noncommutative subspace of Sq2n . 2n ∼ ) = A(Sq2n ), we have another isomorphism Because of the isomorphism A(S1/q 2n−1 ∼ 2n−1 A(S1/q ) = A(Sq ), and again we can assume that |q| > 1 without any loss of generality. Remark 1. The algebras of our spheres, both in even and odd “dimensions”, are generated by the entries of a projections. This is the same as the condition of full projection used by S. Waldmann in his analysis of Morita equivalence of star products [63].

Firstly, if C(S 1 ) ⊗ K ⊆ ker ψ, then ψ factors through C(Sq2n−1 ) and is really a representation of Sq2n−1 . Otherwise, ψ restricts to an irreducible representation of C(S 1 ) ⊗ K. This factorizes as the tensor product of an irreducible representation of C(S 1 ) with one of K. The irreducible representations of C(S 1 ) are simply given by the points of S 1 , and as we have mentioned, K has a unique irreducible representation. The representations of C(S 1 ) ⊗ K are thus classiﬁed by the points of S 1 .

The other generator of K 0 (S 2n ) is the left handed spinor bundle. One K-homology generator [ε] ∈ K0 (S 2n ) is “trivial” and is the pushforward of the generator of K0 (∗) ∼ = Z by the inclusion ι : ∗ → S 2n of a point (any point) into the sphere. The other generator, [µ] ∈ K0 (S 2n ), is the Korientation of S 2n given by its structure as a spin manifold [10]. 46 G. Landi For an odd dimensional sphere, the groups are K 0 (S 2n+1 ) ∼ =Z, 2n+1 ∼ )=Z, K0 (S K 1 (S 2n+1 ) ∼ =Z, 2n+1 ∼ K1 (S )=Z. The generator [1] ∈ K 0 (S 2n+1 ) is the trivial 1-dimensional bundle.