  We introduce a class of operators associated with the signature of a smooth
path $X$ with values in a $C^{\star}$ algebra $\mathcal{A}$. These operators
appear naturally in Taylor expansions of solutions to controlled differential
equations of interest in non-commutative probability theory. They are built by
fully -- and partially -- contracting iterated integrals of $X$, seen as
tensors, with the product of $\mathcal{A}$. We explain how these operators
yield a trajectory on a convolution group of representations of a Hopf monoid
of leveled forests we introduce. Finally, if considering only full contraction
operators we build a trajectory on a group of triangular operators, thereby
obtaining a notion of signature for a path, relevant for non-commutative
probability.
