By J. Weeks

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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 ). Then, αβ αβ ∩ij = [pi ], [pj ] = tr(γλ(pi )ρ(pj )) ⎛ ⎞ λα (pi ) ⊗ γαβ ⊗ λβ (pj )⎠ = tr ⎝ b α,β∈A odd tr(λα (pi )) tr(λβ (pj ))(meven αβ − mαβ ) = b α,β∈A N τi τj (meven ij − modd ij ).

Let ULR ) act on L1A (Heven , Hodd ; J) by A (H ˜ J˜∗ ΔU ∗ (U, Δ) → JU even for U ∈ ULR ) and Δ ∈ L1A (Heven , Hodd ; J). Then the map A (H even ) D(A, H, γ, J) → L1A (Heven , Hodd ; J)/ ULR A (H deﬁned by [D] → [P odd DP even ] is a homeomorphism. even even ) on LR , Hodd ). 25. The map Rn : LR , Hodd ) → L1A (Heven , Hodd ; J) A (H ˜ ∗ J˜ is a surjection intertwining the actions of the deﬁned by Rn (M ) := M + εJM LR even R even group UA (H ) on LA (H , Hodd ) and L1A (Heven , Hodd ; J), and ker(Rn ) ⊂ LR even odd , H ).

Let (H, γ) be an orientable A-bimodule. 19) N D= λ(ei )[D, λ(ej )] + i,j=1 i=j ρ(ek )[D, ρ(el )]. k,l=1 k=l Proof. Fix D ∈ D0 (A, H, γ), and let N N T := D − λ(ei )[D, λ(ej )] − i,j=1 i=j N =D− ρ(ek )[D, ρ(el )] k,l=1 k=l N λ(ei )Dλ(ej ) − i,j=1 i=j ρ(ek )Dρ(el ). k,l=1 k=l Then for all α, β, γ, δ ∈ A, ⎧ γδ ⎪ ⎨Dαβ γδ γδ Tαβ = −Dαβ ⎪ ⎩ 0 if r(α) = r(γ), r(β) = r(δ), if r(α) = r(γ), r(β) = r(δ), otherwise, where for α ∈ A, r(α) is the value of j ∈ {1, . . , N } such that α ∈ Mkj (Kj ). 13, Dαβ must vanish in the ﬁrst, so that T = 0.