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.

中文翻译：

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为局部紧致量子群生成泛函

局部紧量子群上状态的卷积半群的每个对称生成泛函都被证明允许在其域中具有类似核的特性的稠密单位 $*$-子代数。另一方面，我们证明，满足特定技术条件的局部紧量子群的通用 C$^*$-代数单元化的稠密 $*$-子代数上的每个归一化、对称、厄密条件正泛函，扩展以规范的方式生成函数。概述了这些结果的一些后果，特别是那些与从卷积半群构造 cocycles 相关的结果。