Star-cumulants of free unitary Brownian motion

We study joint free cumulants of utut and ut⁎, where utut is a free unitary Brownian motion at time t  . We determine explicitly some special families of such cumulants. On the other hand, for a general joint cumulant of utut and ut⁎, we “calculate the derivative” for t→∞t→∞, when utut approaches a Haar unitary. In connection to the latter calculation we put into evidence an “infinitesimal determining sequence” which naturally accompanies an arbitrary R-diagonal element in a tracial ⁎-probability space.