  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
serve as the basis of Taylor expansions of solutions to controlled differential
equations of interest in noncommutative probability. They are defined by fully
contracting iterated integrals of $X$, seen as tensors, with the product of
$\mathcal{A}$. Were it considered that partial contractions should be included,
we explain how these operators yield a trajectory on a group of representations
of a combinatorial Hopf monoid. To clarify the role of partial contractions, we
build an alternative group-valued trajectory whose increments embody
full-contractions operators alone. We obtain therefore a notion of signature,
which seems more appropriate for noncommutative probability.
