Generating functionals for locally compact quantum groups

Abstract

Every symmetric generating functional of a convolution semigroup of states on a locally compact quantum group is shown to admit a dense unital $\ast$-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 $\ast$-subalgebra of the unitisation of the universal C${}^{\ast }$-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.

25 pages. v2: added an example and made several minor changes. To appear in International Mathematics Research Notices