Sums of Commutators in Free Probability

Abstract

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.

21 pages; numerous improvements suggested by an anonymous referee; figures changed to black and white (color can be switched on in latex file); limit theorems moved to separate paper arXiv:2004.02679

References in corpus (5)

• The Trace Method for Cotangent Sums
• Higher-order tangent and secant numbers
• The Free Tangent Law
• Supports, regularity, and $⊞$-infinite divisibility for measures of the form $({\mu }^{⊞p}{)}^{\uplus q}$
• Inevitable Dottie Number. Iterals of cosine and sine

Cited by in corpus (4)

• The Trace Method for Cotangent Sums
• The Free Tangent Law
• Universality of free random variables: atoms for non-commutative rational functions
• On the anti-commutator of two free random variables