Odd characteristic classes in entire cyclic homology and equivariant loop space homology

Given a compact manifoldM and a smooth map g:M → U.(l×l:C) from M to the Lie group of unitary l×l matrices with entries in C, we construct a Chern character Ch-(g) which lives in the odd part of the equivariant (entire) cyclic Chen-normalized cyclic complex Nϵ(ωT(M × T)) of M, and which is mapped to the odd Bismut-Chern character under the equivariant Chen integral map. It is also shown that the assignment g → Ch-(g) induces a well-defined group homomorphism from the K-1 theory of M to the odd homology group of Nϵ(ωT(M × T)).