Integration with respect to the non-commutative fractional Brownian motion

We study the issue of integration with respect to the non-commutative fractional Brownian motion, that is the analog of the standard fractional Brownian in a non-commutative probability setting.When the Hurst index $H$ of the process is stricly larger than $1\mathrm{/}2$, integration can be handled through the so-called Young procedure. The situation where $H=1\mathrm{/}2$ corresponds to the specific free case, for which an It{^o}-type approach is known to be possible.When $H<1\mathrm{/}2$, rough-path-type techniques must come into the picture, which, from a theoretical point of view, involves the use of some a-priori-defined L{'e}vy area process. We show that such an object can indeed be \enquote{canonically} constructed for any $H\in (\frac{1}{4},\frac{1}{2})$. Finally, when $H\le 1\mathrm{/}4$, we exhibit a similar non-convergence phenomenon as for the non-diagonal entries of the (classical) L{'e}vy area above the standard fractional Brownian.