Generating Functionals for Locally Compact Quantum Groups

Every symmetric generating functional of a convolution semigroup of states on a locally compact quantum group is shown to admit a dense unital ∗-subalgebra with core-like properties in its domain. On the other hand we prove that every normalised, symmetric, hermitian conditionally positive functional on a dense ∗-subalgebra of the unitisation of the universal C$^∗$-algebra of a locally compact quantum group, satisfying certain technical conditions, extends in a canonical way to a generating functional. Some consequences of these results are outlined, notably those related to constructing cocycles out of convolution semigroups.