The odd Chern character in cyclic homology and spectral flow (Q1312066)

Let \({\mathcal A}\) be a Banach \(*\)-algebra. Then the spectral flow defines a pairing \(\langle D,g \rangle : K^{-1} ({\mathcal A}) \times K_ 1 ({\mathcal A}) \to \mathbb{Z}\) with integer values, pairing on odd theta-summable Fredholm module \(D\) over \({\mathcal A}\) representing a class \([D]\) in \(K^{- 1} ({\mathcal A})\) with a unitary matrix \(g\) with entries in \({\mathcal A}\) representing an element in \(K_ 1({\mathcal A})\). In his main result, the author expresses \(\langle D,g \rangle\) in terms of the Chern character \(Ch^*(D)\) of \(D\) in entire cyclic cohomology introduced by \textit{A. Jaffe}, \textit{A. Lesniewski} and \textit{K. Osterwalder} [Commun. Math. Phys. 118, No. 1, 1-14 (1988; Zbl 0656.58048)] and the Chern character \(Ch_ * (g)\) of \(g\) in entire cyclic homology. Precisely, he proves the formula \(\langle D,g \rangle = (Ch^* (D),Ch_ * (g))\).