Transgression of the Chern character for families and cyclic cohomology

The aim of this note is to establish a link between the Chern character in K-homology [3] and the version of the standard Chern character and of its transgression developed by D. Quillen [5] and J.-M. Bismut [2]. We show that the Chern character in K-homology, which associates to every finitely summable Fredholm module over an involutive algebra a cyclic cohomology class, is equal via the Loday-Quillen isomorphism t