Convolution semigroups on locally compact quantum groups and noncommutative Dirichlet forms

Adam Skalski, Ami Viselter

Research output: Contribution to journalArticlepeer-review


The subject of this paper is the study of convolution semigroups of states on a locally compact quantum group, generalising classical families of distributions of a Lévy process on a locally compact group. In particular a definitive one-to-one correspondence between symmetric convolution semigroups of states and noncommutative Dirichlet forms satisfying the natural translation invariance property is established, extending earlier partial results and providing a powerful tool to analyse such semigroups. This is then applied to provide new characterisations of the Haagerup Property and Property (T) for locally compact quantum groups, and some examples are presented. The proofs of the main theorems require developing certain general results concerning Haagerup's L p -spaces.

Original languageEnglish
Pages (from-to)59-105
Number of pages47
JournalJournal des Mathematiques Pures et Appliquees
StatePublished - Apr 2019

Bibliographical note

Funding Information:
This author was partially supported by the National Science Centre (NCN) grant no. 2014/14/E/ST1/00525.

Publisher Copyright:
© 2018 Elsevier Masson SAS


  • Convolution operator
  • Convolution semigroup
  • Locally compact quantum group
  • Noncommutative Dirichlet form

ASJC Scopus subject areas

  • Mathematics (all)
  • Applied Mathematics


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