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One then has that for any gauge potential A, (A, H, D, J, γ) and (A, H, DA , J, γ) are Morita equivalent. 14. Let (H, J, γ) be an orientable real A-bimodule of even KOdimension. 20) λ(ei )[D, λ(ej )] i,j=1 i=j is a gauge potential for the real spectral triple (A, H, D, J, γ) such that DA = 0. Thus, every ﬁnite orientable real spectral triple (A, H, D, J, γ) of even KOdimension is Morita equivalent to the dynamically trivial triple (A, H, 0, J, γ). 4. Real spectral triples of odd KO-dimension. For this section, let (H, J) be a real A-bimodule of odd KO-dimension n mod 8 with multiplicity matrix m.

9) tr(λα (pi )) = τi 0 if α ∈ Mni (Ki ), otherwise. Now, if (H, γ) is an even A-bimodule with intersection form ·, · , we can deﬁne a matrix ∩ ∈ MN (Z) by ∩ij := [pi ], [pj ] . e. (H, γ) satisﬁes Poincar´e duality) if and only if ∩ is non-degenerate. 4]). Let (H, γ) be an even A-bimodule with pair of multiplicity matrices (meven , modd ). 11) ∩ij = τi τj meven ij − modd ij , so that (H, γ) satisﬁes Poincar´e duality if and only if the matrix meven − modd is non-degenerate. Proof. First, since H = Heven ⊕ Hodd , we can write 1nα ⊗ γαβ ⊗ 1nβ , γ= b α,β∈A FINITE SPECTRAL TRIPLES 27 where γαβ = 1meven ⊕ (−1modd ).

1. KO-dimension 0 or 4 mod 8. e. of KO-dimension 0 or 4 mod 8. Let (H, γ, J) be a real A-bimodule of KO-dimension n mod 8, for n = 0 or 4; let the mutually orthogonal projections P even and P odd on H be deﬁned as before. Then, since [J, γ] = 0, we have that J = J even ⊕ J odd , where J even = P even JP even and J odd = P odd JP odd . One can then check that (Heven , J even ) and (Hodd , J odd ) are real A-bimodules of KO-dimension 1 or 7 mod 8 if n = 0, and 3 or 5 mod 8 if n = 4. On the other hand, given (Heven , J even ) and (Hodd , J odd ), one can immediately reconstruct (H, γ, J) by setting γ = 1Heven ⊕ (−1Hodd ) and J = J even ⊕ J odd .