Sums of commutators in free probability

Wiktor Ejsmont*, Franz Lehner

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


We study the linear span of commutators of free random variables and show that these are the only quadratic forms which satisfy the following equivalent properties: • preservation free infinite divisibility, • free and strong cancellation of odd cumulants, • symmetric distribution for any free family. The main combinatorial tool is an involution on non-crossing partitions.

Original languageEnglish
Article number108791
JournalJournal of Functional Analysis
Issue number2
Publication statusPublished - 15 Jan 2021


  • Cancellation of free cumulants
  • Commutator
  • Free infinite divisibility

ASJC Scopus subject areas

  • Analysis


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