On the multiplication of operator-valued c-free random variables

We discuss some results concerning the multiplication of non-commutative random variables that are c-free with respect to a pair $(\mathrm{\Phi },\phi )$, where $\mathrm{\Phi }$ is a linear map with values in some Banach or C${}^{\ast }$-algebra and $\phi$ is scalar-valued. In particular, we construct a suitable analogue of the Voiculescu’s $S$-transform for this framework.