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Given a compact manifold $M$ and $g\in C^{\infty}(M,U(l;\mathbb{C}))$ weconstruct a Chern character $\mathrm{Ch}^-(g)$ which lives in the odd part ofthe equivariant (entire) cyclic Chen-normalized bar complex$\underline{\mathscr{C}}(\Omega_{\mathbb{T}}(M\times \mathbb{T}))$ of $M$, andwhich is mapped to the odd Bismut-Chern character under the equivariant Chenintegral map. It is also shown that the assignment $g\mapsto \mathrm{Ch}^-(g)$induces a well-defined group homomorphism from the $K^{-1}$ theory of $M$ tothe odd homology group of $\underline{\mathscr{C}}(\Omega_{\mathbb{T}}(M\times\mathbb{T}))$

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