  We calculate for the first time the shear viscosity of ultrarelativistic
Boson systems in the presence of a Bose-Einstein condensate (BEC). Two
different methods are used. One is the Grad's method of moments and another is
the Green-Kubo relation within a kinetic transport approach. In this work we
consider a Boson system with isotropic elastic collisions and a gluon system
with elastic scatterings described by perturbation QCD (pQCD). The results show
that the presence of a BEC lowers the shear viscosity. This effect becomes
stronger for the increasing proportion of the BEC in the Boson system and is
insensitive to the detail of interactions.
