  We study states on the universal noncommutative *-algebra generated by the
coefficients of a unitary matrix, or equivalently states on the unitary dual
group. Its structure of dual group in the sense of Voiculescu allows to define
five natural convolutions. We prove that there exists no Haar state for those
convolutions. However, we prove that there exists a weaker form of absorbing
state, that we call Haar trace, for the free and the tensor convolutions. We
show that the free Haar trace is the limit in distribution of the blocks of a
Haar unitary matrix when the dimension tends to infinity. Finally, we study a
particular class of free L\'evy processes on the unitary dual group which are
also the limit of the blocks of random matrices on the classical unitary group
when the dimension tends to infinity.
