Regular C∗-valued weights

We introduce the notion of a C *-valued weight between two C *-algebras as a generalization of an ordinary weight on a C *-algebra and as a C *-version of operator valued weights on von Neumann algebras. Also, some form of lower semi-continuity will be discussed together with an extension to the Multiplier algebra. A strong but useful condition for C *-valued weights, the so-called regularity, is introduced. At the same time, we propose a construction procedure for such regular C *-valued weights. This construction procedure will be used to define the tensor product of regular C *-valued weights.