On Additive K-Theory with the Loday - Quillen *-Product

Abstract

The $*$-product defined by Loday and Quillen [17] on the additive $\mathbf{K}$-theory (the cyclic homology with shifted degrees) $K_*^+(A)$ for a commutative ring $A$ is naturally extended to a product ($*$-product) on the additive $\mathbf{K}$-theory $K_*^+(\Omega)$ for a differential graded algebra $(\Omega,d)$ over a commutative ring. We prove that Connes' $\mathbf{B}$-maps from the additive $\mathbf{K}$-theory $K_*^+(\Omega)$ to the negative cyclic homology $\mathrm{HC}_*^-(\Omega)$ and to the Hochschild homology $\mathrm{HH}_*(\Omega)$ are morphisms of algebras under the $*$-product on $K_*^+(\Omega)$. Applications to topology of Connes' $\mathbf{B}$-maps are also described.