The Adams operation acts on the algebraic $K$-theory of $R$ but the action as far as I know doesn't come from a endo-functor on the category of projective modules over $R$. For the $K_0$ there is an easy way of defining it by using the splitting of the vector bundles by a pullback and defining it over split vector bundles. This assignment is not well-defined at the level of vector bundles but it becomes a well defined ring homomorphism on the level of $K_0$. So I have the following questions:
Is it possible to define the Adams operation on the rational homology of the $K$-theory spectrum. (You can take either the plus construction which the homology agrees with the homology of $BGL_n(R)$ or the $Q$-construction without taking the loop space preferably.)? The primitive elements of the rational homology agrees with the rational homotopy so it should be possible to define on them. What about the non-primitive ones? If it is possible how does it react to the product on the homology (coming from being an $H$-space)? Well is there any nice description of this action if they exist?