  For a Green biset functor $A$, we define the commutant and the center of $A$
and we study some of their properties and their relationship. This leads in
particular to the main application of these constructions: the possibility of
splitting the category of $A$-modules as a direct product of smaller abelian
categories. We give explicit examples of such decompositions for some classical
shifted representation functors. These constructions are inspired by similar
ones for Mackey functors for a fixed finite group.
