  We study different possibilities to apply the principles of rough paths
theory in a non-commutative probability setting. First, we extend previous
results obtained by Capitaine, Donati-Martin and Victoir in Lyons' original
formulation of rough paths theory. Then we settle the bases of an alternative
non-commutative integration procedure, in the spirit of Gubinelli's controlled
paths theory, and which allows us to revisit the constructions of Biane and
Speicher in the free Brownian case. New approximation results are also derived
from the strategy.

Probability;Operator Algebras