On Additive K-Theory with the Loday - Quillen *-Product - Kuribayashi, Katsuhiko; Yamaguchi, Toshihiro (Mathematica Scandinavica, 2000)
The $*$-product defined by Loday and Quillen [17] on the additive $K$-theory (the cyclic homology with shifted degrees) $K_*^+(A)$ for a commutative ring $A$ is naturally extended to a product ($*$-product) on the additive $K$-theory $K_*^+(Ω)$ for a differential graded algebra $(Ω,d)$ over a commutative ring. We prove that Connes' $B$-maps from the additive $K$-theory $K_*^+(Ω)$ to the negative cyclic homology $HC_*^-(Ω)$ and to the Hochschild homology $HH_*(Ω)$ are morphisms of algebras under the $*$-product on $K_*^+(Ω)$. Applications to topology of Connes' $B$-maps are also described.