Caractère multiplicatif d'un module de Fredholm. (Multiplicative character of a Fredholm module)
Let A be an algebra. An odd Fredholm module is an action \(\rho\) of A on a \({\mathbb{Z}}/2\)-graded Hilbert space \(H=H^+\oplus H^-\) such that the off- diagonal elements of \(\rho\) (a), \(a\in A\) are compact. An even Fredholm module consists of two actions \(\rho_ 1\), \(\rho_ 2\) of A on a Hilbert space such that \(\rho_ 1(a)-\rho_ 2(a)\), \(a\in A\) is compact. If the off-diagonal elements or these differences are not only compact but even in the Schatten ideal \(\ell^ p\) one says that the module is p-summable. The module is said to be of dimension p if it is \((p+1)\)-summable and of the same parity as p. It is well known (by previous results of the first author) that any Fredholm module of dimension p induces additive homomorphisms \(K_ i(A)\to {\mathbb{Z}}\subset {\mathbb{C}}\) where \(i=0,1\) according to the parity of p. In the present article the authors show that such a module also induces a multiplicative homomorphism \(K_{p+1}(A)\to {\mathbb{C}}^*\) where \(K_ n\) is the algebraic K-theory in the sense of Quillen. It suffices, obviously, to construct this homomorphism for the universal algebra \({\mathcal M}^ p\) connected with a universal Fredholm module of dimension p. This algebra is a Banach algebra and one obtains the desired homomorphism from an exact sequence connecting the algebraic and topological K-theory using certain canonical cyclic cocycles on \({\mathcal M}^ p.\) Finally the authors describe the central extensions of SL(\({\mathcal C}^{\infty}(S^ 1))\) and of \(Diff^+(S^ 1)\) associated to certain Fredholm modules and their relation to the Virasoro algebra.
algebraic K-theory in the sense of Quillen
universal Fredholm module of dimension p
scientific article; zbMATH DE number 4098061
scientific article