the q-circular system is a tuple of non-commutative random variables (operators with some state) which interpolate independent standard complex gaussian random variables (q=1) in classical probability and freely independent circular random variables (q=0) in free probability. one of the interesting results on q-deformed probability is that -1<q<1 case has similar properties to free case (q=0). haagerup inequality is one of such properties, which was originally proved for generators of free groups with respect to the left regular representation.    

 in this talk, i will explain the strong version of haagerup inequality for the q-circular system, which was originally proved by kemp and speicher for q=0. this talk is based on a joint project with t. kemp.