# Generating functionals for locally compact quantum groups

## TL;DR

This paper explores the structure of generating functionals in locally compact quantum groups, showing how symmetric functionals can be densely approximated and extended, with implications for constructing cocycles from convolution semigroups.

## Contribution

It establishes core-like properties for symmetric generating functionals and provides a method to extend certain positive functionals to generating functionals in quantum groups.

## Key findings

- Symmetric generating functionals admit dense subalgebras with core properties.
- Certain positive functionals extend canonically to generating functionals.
- Results facilitate the construction of cocycles from convolution semigroups.

## Abstract

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.