The authors gave results of cohomological informations related to a non-tracial weight \(\Phi\) on a C*-algebra \(A\). Assume that \(\Phi\) has the \(KMS_\beta\) property with respect to a one-parameter group of automorphism \(\sigma\) of \(A\) and that \(\sigma: \mathbb R \to \text{Aut}(A)\) factorizes through a circle action \(\sigma: \mathbb R \to \text{Aut}(A)\). The element \(\psi = (b+B)\Phi\) is a cocycle on the quotient complex \((CC_{\text{ev}}(A)/CC_{\text{ev}}(A^\sigma), b+B)\) and gives a pairing with the \(K\)-theory of the mapping cone \(M\) of the inclusion \(A^\sigma \hookrightarrow A\). The action of \(\sigma\) induces an element \([\hat{\mathcal D}]\) of the KK-group \(KK^{\mathbb T}_0(M,A^\sigma)\) and the Kasparov product gives the homomorphism
\[ [\hat{\mathcal D}] \cap : K^{\mathbb T}_0(M) \to K^{\mathbb T}_0(A^\sigma)= K_0(A^\sigma)[\chi, \chi^{-1}], \]
where \(\chi\) is the fundamental character of \(\mathbb T\). The pairing of \([\hat{\mathcal D}]\) with \(K^{\mathbb T}_0(M)\) is given in Theorem 2.11: Let \([v]\) be a class in \(K^{\mathbb T}_0(M)\) represented by a partial isometry \(v\) in \(\left(A^\sim \otimes B(\mathcal H) \right)^{\mathbb T}\), then
\[ \text{Index}_{\hat{\mathcal D}}([v]) = -\text{Index}\left((P \otimes 1)v(P\otimes 1): v^*v(P\mathcal H_F \otimes H) \to vv^*(P\mathcal H_F \otimes \mathcal H)\right). \]
For a faithful, positive, semifinite weight \(\phi\) on a C*-algebra \(A\) satisfying the \(KMS_\beta\) condition, with respect to the action of \(\mathbb T\) induced by \(\sigma\), denote \(\mathcal H_\phi = L^2(A,\phi)\), \(\pi_\phi: A \to \mathcal B(\mathcal H_\phi)\) the GNS-representation, \(J_\phi\) the conjugate linear isometry on \(\mathcal H_\phi\), \(\pi_\phi(A)' = J_\phi\overline{\phi_\phi(A)}J_\phi\), and denote \(\mathcal N = J_\phi\overline{\phi_\phi(A^\sigma)}J_\phi\).
Identifying a natural faithful semifinite weight \(\phi\) on the C*-algebra \(\mathcal N\) with a semitrace \(Tr_\phi\) we study the spectral flow of a path of operators of the form \(t \in [0,1] \mapsto \mathcal D + a_t\), where \(\mathcal D\) is the generator of the action of \(\mathbb T\) and \(a_t\) is a path of bounded perturbation associated to a class in \(K^{\mathbb T}_0(M)\).
If \((A,\pi_\phi,\mathcal H_\phi,\mathcal D)\) is a spectral triple with respect to \((\mathcal N, Tr_\phi)\), the semifinite von Neumann algebra \(\mathcal M = \mathcal N_\phi\) and the trace \(\phi_{\mathcal D} = Tr_\phi(e^{-\beta\mathcal D/2}\cdot e^{-\beta\mathcal D/2})\) is a trace on \(\mathcal M\) and one makes a modular index pairing by an analytical method (Theorems 5.6, 4.10, 5.9, and also 1.2, 1.3, 1.4). This puts the previous considered example of Cuntz algebras and \(SU_q(2)\) in a general framework and then the authors consider a new example of Araki-Woods \(III_\lambda\) representations of the Fermion algebra.